How to Allocate More Space to a Filesystem Mounted on /

First, don't do anything destructive until you understand exactly what's going on. For example, don't run e2fsck without -n on a device unless you're absolutely sure that the device contains an ext2/ext3/ext4 filesystem and that you're in a position to recover it.

Before risking anything destructive, I recommend that you make a copy of the raw disks. Get a disk that's at least as large and copy the whole thing:

cat </dev/sdb >/dev/sdd

(where the disk to save is /dev/sdb and you want to overwrite /dev/sdd — be very careful not to get a single letter wrong!).

/dev/sdb seems to be in an inconsistent state. Apparently it has a partition table that indicates that there is a partition /dev/sdb1, and at the same time it has a filesystem on the whole disk rather than on the partition. If there is indeed a filesystem on the whole disk, there shouldn't be a partition table. Given that you've already done something potentially destructive, I recommend you try copying all the files off the disk. Mount it as read-only:

mkdir /media/b
mount -r -t ext4 /dev/sdb /b
cp -a /media/b /some/place/you/have/enough/space

/dev/sdc1 contains an LVM physical volume. You need to assemble this volume. In fact, the system may have done that already; the data is on the logical volume, not directly on the partition. Run lvs to see what logical volumes are available.

mkdir /media/c
lvs   # You'll probably see a line with "  volumegroupname    logicalvolumename"
mount -r -t ext4 /dev/mapper/volumegroupname-logicalvolumename /media/c

If lvs doesn't show anything, you'll need to activate the volumes first. Run vgs to see a list of available volume groups, vgchange -ay volumegroupname to activate a volume group, lvchange -ay volumegroupname/logicalvolumename to activate a logical volume. If you need help, post the output of pvs, vgs and lvs.

You can try to recover the filesystem structure even if e2fsck doesn't manage. But this requires deep knowledge of the filesystem. Even a professional may or may not get far, depending on precisely which bits were overwritten. TestDisk is a filesystem recovery tool; try unleashing it on your disk.

If you don't manage to repair the filesystem, go fishing for individual files. This may be easy or difficult depending on what kinds of files you're after and how the disk is laid out. A favorable case is large video files that were all written in one go (so each file has a recognizable header and is likely to be unfragmented). An unfavorable case is a tree of source code where you'll have a lot of versions of each file (all but one deleted) and even associating file data with names is part of the difficulty.

There are tools that go looking for recognizable files on a damaged filesystem (or erased files), mostly images and videos and sounds. The TestDisk website has a list of data recovery distributions; CAINE, in particular, is a live distribution specialized in forensics.

Latecomer here. It may be a little counterintuitive, but I use the service (rather than mount) systemd user unit and it works for me. I had to add the user and noauto options to /etc/fstab entry.

cat ~/.config/systemd/user/mount@.service
[Unit]
Requires=home-me.mount
After=home-me.mount

[Service]
ExecStart=/bin/mount %h/%I
ExecStop=/bin/umount %h/%I
RemainAfterExit=yes

[Install]
WantedBy=default.target

You should enable the unit instance with a command such as:

systemctl --user enable mount@some-directory

Help with the @ in the filename, can be found reading about systemd instantiated units.