I get a drive letter just fine under XP. Under XP a dialog pops up each time I plug it in, asking how it should be handled. By canceling it ends up being handled as a drive.
My guess is that there's a setting, probably related to cameras and photos, that will change how Windows 7 handles it by default. Unfortunately I've no Win 7 machine handy.
A couple of suggestions from Googling for answers:
Many people have solved the problem by deleting any photos on their iPhone that weren't actually taken with the iPhone. Anything received in email, saved to disk in Safari, etc. After clearing those out the device again appeared as a drive.
Reinstall the iPhone driver in Windows 7 by following instructions from Apple Support.
EDITED TO ADD WIN XP, AS REQUESTED
Getting Windows XP to Present Itself as a Drive-Like Camera
First, the way I have things set up, which may or may not affect how it works for you: I am running all of the XP service packs, updated regularly through Software Update, with no special drivers or applications. My account is an administrator on my PC. iTunes 10.1 is installed, though it is not involved with the camera stuff (it's possible that having iTunes installed does provide some kind of driver, I don't know). My iPhone is a stock iPhone 3GS, non-jailbroken, running iOS 4.2.1. Note that all of this worked identically with iTunes 9 and my phone running iOS 3.0.1.
Event Handling
The first time you attach your iPhone to your WinXP PC a dialog much like the following should appear:
Drive List
If you choose one of the Microsoft Office handlers, the Wizard, or another app like Photoshop, your phone will not present itself as a camera (more on changing that behavior below). If you click Cancel then the dialog will disappear and the camera will become available.
Windows Explorer Media Directory
If you then double-click on the iPhone icon all of the images and movies are presented:
You can freely drag photos and videos out of the directory. You can drag stuff in, too.
Changing the Event Handler
If at some point you chose an event handler rather than clicking Cancel you can change how XP handles the iPhone. You'll need to get to the phone's Properties, which may require opening the Scanners and Cameras control panel. Click on the dialog's Events tab to see this:
If you change your current setting to Prompt then you'll get that first dialog I showed when you plug in your device. You can also choose Take no action if you want it to always behave like a drive.
This works well for iOS 10 and macOS 10.12:
You're not the first to ask this question. Searching with Google shows that it's been discussed here and here.
Unfortunately, iOS 9 and earlier do not support what you're describing.
You mention that you don't have a wireless card but perhaps you can share your internet connection wirelessly, through an Ad-Hoc network. I think that's the best way to go, honestly.
This doesn't work for internet:
[PC] --->--(USB)--->--> [iOS Device]
But this probably would, since the iOS device doesn't care, for the most part, where the WiFi is coming from:
[PC] --->--(WiFi)--->--> [iOS Device]
So, you might consider grabbing a wireless card for your PC. Alternatively, a jailbroken iPhone might be able to get more mileage with a USB cable than a vanilla one would.
Best Answer
Generally speaking you can't do this on a PC (Linux or Windows) or a Mac. Though on Windows PCs you can usually get to the folder that holds music and pictures, if I remember correctly.
But really there is no way to treat an iPhone like a thumb drive without jailbreaking it or jumping through hoops that would end up (possibly) being more trouble than it is worth.
The only thing you should do is back up the phone to your PC from time to time. Yes you can use iCloud backup, which I do use, but I also change that button in iTunes and back up locally from time to time. Mostly because I am a vaguely paranoid IT guy who has seen so many people lose data because of incomplete backups that I have two backup methods for my iPhone.