Short Answer
The drivers you found are old versions:
$ dpkg -l | grep Brother
ii brother-cups-wrapper-common 1.0.0-10-0ubuntu6 amd64 Common files for Brother cups wrapper packages
ii brother-udev-rule-type1 1.0.0-1 all Brother udev rule type 1
ii brscan-skey 0.2.4-1 amd64 Brother Linux scanner S-KEY tool
ii brscan2 0.2.5-1 amd64 Brother Scanner Driver
ii printer-driver-brlaser 3-5~ubuntu1 amd64 printer driver for (some) Brother laser printers
ii printer-driver-ptouch 1.4-1 amd64 printer driver Brother P-touch label printers
The newest versions are:
$ dpkg -l | grep -i brother
ii brother-udev-rule-type1 1.0.2 all Brother udev rule type 1
ii brscan-skey 0.2.4-1 amd64 Brother Linux scanner S-KEY tool
ii brscan4 0.4.4-3 amd64 Brother Scanner Driver
ii cupswrapperdcp7065dn:i386 2.0.4-2 i386 Brother DCP7065DN CUPS wrapper driver
ii dcp7065dnlpr:i386 2.1.0-1 i386 Brother DCP-7065DN LPR driver
ii printer-driver-brlaser 3-5~ubuntu1 amd64 printer driver for (some) Brother laser printers
ii printer-driver-ptouch 1.4-1 amd64 printer driver Brother P-touch label printers
udev
rule upgraded from 1.0.0-1
to 1.0.2
brscan2
version 0.2.5-1
upgraded to brscan4
version 0.4.4-3
Also you have a generic cups-wrapper
where as mine is model specific.
TL:DR; Very, very LONG ANSWER
Much of the long answer is trial and error. You need only follow Step 1, Step 2, the first part of Step 3 (stop at selecting scanner only). Then skip down to Step 6 and Step 7.
Background
I have a Brother DCP-7065DN which is a nice monochrome double sided laser printer. I've had it for three years but have never used the scanner feature primarily because "whose got paper to scan in an electronic world?". Due to your question I thought it was time to get the scanner function working.
Comparing DCP-7065DN to DCP-350C
According to this post in Ubuntu Forums the same Brother Driver tool works for both your DCP-350C and my DCP-7065DN printer.
Current installation tools
Your question contains no less than 15 links and I noticed the site of your first link is out-dated:

The Ubuntu Forums Q&A I linked above contains a more current installation tool link:

Notice the Release Date is Christmas 2017, only a few months old.
Rather than repeating your steps I'll start a fresh install (of the scanner only) using the Ubuntu Forums link above.
Installation Step 1 - Select Printer
Start by selecting the right product. In the printer icon for MFC-J410W
, click the link Not your product? and a selection page appears:

Your printer can be found in the first icon which has a sub-menu for "DCP 3 Series" which leads to your specific model:

My printer is the fourth icon in the panel, which being 3 icons wide is immediately below yours. I'll select that and it has a sub-menu for "DCP 7 Series" which after clicking presents:

I'll click on DCP-7065DN. You would have already selected DCP-350C on the previous screen shot.
Installation Step 2 - Select Operating System (OS)
After selecting the printer model you are asked for your OS. It defaults to Linux and ".rpm". The default is ".rpm" which stands for Red Hat or Arch Linux or something like that. Change the distro to ".deb" which stands Debian which is really Ubuntu.

Click the Search
button.
Installation Step 3 - Download Software
Now that the preliminaries are finished and the Command Line aficionados and GUI haters in Ubuntu-Verse have put my name on their dart boards we get to select the software to download:

The model icon in the top-left corner should be DCP-350C for you but it is DCP-7065DN for me.
The first section is for printer drivers but we don't need that as both our printers print fine. The last section is for source code but we don't need that. If we did we probably wouldn't be on this website. The middle section contains the scanner drivers:

Here we need to select the architecture: 32-bit or 64-bit.
I've read a 32-bit driver won't work on my 64-bit arch so I'm only going to download #3 & #4 for now. I may come back and grab #1 if I need the driver?
Installation Step 3-A)
Download Scan-key-tool 64bit (deb package)
After selecting #3 scanner option we see your model number on line #2 and my model number on the last line (of the screen shot)...:

...So assuming all works well for me it should all work well for you!
You need to agree to the EULA (End User License Agreement) and then download the .deb
file.
The file downloaded is:
- brscan-skey-0.2.4-1.amd64.deb, size 50.9 KB, date 03/12/2014
The follow-up instructions on the web page says:
brscanX should be installed first
Installation Step 3-B)
Download Scanner driver 32bit (deb package)
So we have to go back a screen and download #1 (**
Scanner driver 32bit (deb package)**). The file downloaded is:
- brscan4-0.4.4-4.i386.deb, size 63.9 KB, date 09/22/2017
Notice the date is only a 6 months old!
Installation Step 3-C)
Download Scanner Setting file (deb package)
At the same time #1 is downloaded, we'll download #4 (Scanner Setting file (deb package)). The file downloaded is:
- brother-udev-rule-type1-1.0.2-0.all.deb, size 1.6 KB, date 11/25/2016
Notice this the version number is different than yours!
Installation Step 4 - sudo dpkg -i ...
To summarize the packages downloaded:
- brscan4-4.4-4.i386.deb
- brscan-skey-0.2.4-1.amd64.deb
- brother-udev-rule-type1-1.0.2.0.all.deb
We will first install #1, then #4, reboot
and finally install #3:
───────────────────────────────────────────────────────────────────────────────
rick@alien:~/Downloads$ sudo dpkg -i brscan4-0.4.4-4.i386.deb
Selecting previously unselected package brscan4:i386.
(Reading database ... 1147453 files and directories currently installed.)
Preparing to unpack brscan4-0.4.4-4.i386.deb ...
Unpacking brscan4:i386 (0.4.4-4) ...
Setting up brscan4:i386 (0.4.4-4) ...
This software is based in part on the work of the Independent JPEG Group.
───────────────────────────────────────────────────────────────────────────────
rick@alien:~/Downloads$ sudo dpkg -i brother-udev-rule-type1-1.0.2-0.all.deb
Selecting previously unselected package brother-udev-rule-type1.
(Reading database ... 1147495 files and directories currently installed.)
Preparing to unpack brother-udev-rule-type1-1.0.2-0.all.deb ...
Unpacking brother-udev-rule-type1 (1.0.2) ...
Setting up brother-udev-rule-type1 (1.0.2) ...
ls: cannot access '/etc/udev/rules.d/*.rules': No such file or directory
───────────────────────────────────────────────────────────────────────────────
rick@alien:~/Downloads$
Ohh No! Error on the last line.
No idea why ls
was generating an error because the directory is there and three rules exist. We can see one of them has just been set up:
$ ll /etc/udev/rules.d/*brother*.rules
-rw-r--r-- 1 root root 1004 Mar 10 16:59 /etc/udev/rules.d/60-brother-libsane-type1.rules
Oh well... let's proceed anyway. Looking at the new Brother udev
rule all it does after a whole bunch of code is set:
ENV{libsane_matched}="yes"
Now it's time to post this answer as is and reboot
.
After rebooting let's install the last file:
$ sudo dpkg -i brscan-skey-0.2.4-1.amd64.deb
Selecting previously unselected package brscan-skey.
(Reading database ... 1147499 files and directories currently installed.)
Preparing to unpack brscan-skey-0.2.4-1.amd64.deb ...
Unpacking brscan-skey (0.2.4-1) ...
Setting up brscan-skey (0.2.4-1) ...
Interesting that there is more than a million files on my system. Those 25 kernels add up over time.
Installation Step 5 - Testing
I cheated a bit and found this: Brother DCP-7065DN, Ubuntu 14.04 64bit: Can print to network printer but cannot use the scan function which recommended these steps:
───────────────────────────────────────────────────────────────────────────────
rick@alien:~/Downloads$ brscan-skey
───────────────────────────────────────────────────────────────────────────────
rick@alien:~/Downloads$ brscan-skey -l
DCP-7065DN : brother4:bus4;dev7 : USB Active
───────────────────────────────────────────────────────────────────────────────
rick@alien:~/Downloads$
So far, so good. The answer (no long sure if it's sure valid) says you need to put brscan-skey
into your Startup Applications and you need gimp
installed. Comment below if you need help doing this.
Apparently xsane
is needed for scanning:
$ sudo apt install xsane
Reading package lists... Done
Building dependency tree
(...BLAH, BLAH, BLAH...)
Setting up xsane-common (0.999-3ubuntu1) ...
Setting up xsane (0.999-3ubuntu1) ...
xsane
= extremely insane
After installing type xsane
in the terminal and get:

After closing the error message, xsane
tries to scan for devices and eventually replies there are none. Oh Joy, Oh Bliss. The Brother website says scanning is designed to be done as root so we'll type sudo xsane
and now we see:

Sounds scary but regular user couldn't find a scanner so let's see if sudo
does. Select Continue at your own risk
button. After window greys out for a few momments the same response as regular user appears: No devices available
.
Software method doesn't work, try hardware method
Using the scan key on the Brother MFC generates this message:

Well at least there is communication between scanner and computer because gimp
started up automatically.
I never rebooted after installing brscan-skey
so do it now out of hope / desperation.
Step 6 - Go back and download CUPS Wrapper
Rebooting didn't work so I'll go back and install the CUPS Wrapper tool dated 12/25/2017
which has not been installed on my system. You have an older version of on your system:

There are five download packages. When downloading the first one (CUPSwrapper) this dialog box appears:

- Change option from
Open
to Save
!
- Click
OK
Open terminal, change to ~/Downloads
and verify file exists:
rick@alien:~/Downloads$ ll *.gz*
-rw-rw-r-- 1 rick rick 24666 Mar 11 08:24 linux-brprinter-installer-2.2.0-1.gz
After downloading these instructions appear on the website. Abbreviated format follows:
rick@alien:~/Downloads$ gunzip linux-brprinter-installer-2.2.0-1.gz
rick@alien:~/Downloads$ sudo su
root@alien:/home/rick/Downloads#
root@alien:/home/rick/Downloads# bash linux-brprinter-installer-2.2.0-1 DCP-7065DN # Change DCP-7065DN to DCP-350C
You are going to install following packages.
dcp7065dnlpr-2.1.0-1.i386.deb
cupswrapperDCP7065DN-2.0.4-2.i386.deb
brscan4-0.4.4-3.amd64.deb
brscan-skey-0.2.4-1.amd64.deb
OK? [y/N] ->
Uh-Oh... It's selecting brscan4-0.4.4-3.amd64.deb
but earlier I had installed brscan4-0.4.4-4.i386.deb
which was a newer version but the wrong architecture. This could explain the errors I had!
Select y
to proceed. Big license agreement pops up. Select y
to proceed. Another license agreement pops up. Select y
to proceed. The script runs apt install
on it's own.
A prompt appears:
Will you specify the Device URI? [Y/n] ->
For yours and mine USB Printer we select n
.
Next it prompts for a test print. To kill another tree select y
. Yeah the test print works and I now have a printed page I can scan later!
A third big license agreement now appears so, select y
to proceed.
It's all finished so reboot
.
Step 7 - Test simple-scan
After reboot
check the installation in terminal and perform simple scan:
───────────────────────────────────────────────────────────────────────────────
rick@alien:~$ brscan-skey
───────────────────────────────────────────────────────────────────────────────
rick@alien:~$ brscan-skey -l
DCP-7065DN : brother4:bus4;dev7 : USB Active
───────────────────────────────────────────────────────────────────────────────
rick@alien:~$ dpkg -l | grep -i brother
ii brother-udev-rule-type1 1.0.2 all Brother udev rule type 1
ii brscan-skey 0.2.4-1 amd64 Brother Linux scanner S-KEY tool
ii brscan4 0.4.4-3 amd64 Brother Scanner Driver
ii cupswrapperdcp7065dn:i386 2.0.4-2 i386 Brother DCP7065DN CUPS wrapper driver
ii dcp7065dnlpr:i386 2.1.0-1 i386 Brother DCP-7065DN LPR driver
ii printer-driver-brlaser 3-5~ubuntu1 amd64 printer driver for (some) Brother laser printers
ii printer-driver-ptouch 1.4-1 amd64 printer driver Brother P-touch label printers
───────────────────────────────────────────────────────────────────────────────
rick@alien:~$ simple-scan
After all the hard work we are rewarded with a scan:

Best Answer
For presently supported devices Brother provide an install script that both, downloads and installs all drivers needed to set up a network printer very quickly.
Set up the network connection
Install the proprietary drivers from Brother
Search for Linux (deb) drivers:
In case we have a locale different from English we may have to ignore a warning to then select Language > English>
Download the Driver Install Tool only (no need to download the other drivers as the installer tool will do that for you).
Unpack the downloaded file and give it executable permission
Run the installer script in a terminal with sudo:
We will be asked to proceed, accept licenses and we will need to answer several questions, e.g.:
Input model name ->
type in the number of your printer (e.g.MFC-J5320DW
)Will you specify the Device URI? [Y/n] ->
SayY
if your printer was installed in the network (n
for USB only).Type in the number of your new printer's URI (e.g.
12
in the example below) from a list of all devices foundChoose to print an Ubuntu print test page with
Y
enter IP address ->
enter here the IP address of your printer noted above to enable network scanning.Find your new printer in the System Settings > Printers.
Install the fax function if available.
Choose the general purpose Brother fax driver for any Brother fax model to download the LPR and CUPS wrapper drivers from their DEB package, e.g.:
These drivers will likely also work with your printer's fax module by adding a
BRFAX
-printer for sending a fax from any application.Note in addition: Some devices will come with a maintenance web-interface we can reach simply by browsing to the printer's IP.