How should I back-up my data before a windows re-install? Omitted files when copying C drive

Just image the whole drive - it'll backup everything, and most decent imaging software should let you do some flavour of compression. You can then mount the drive directly and get files off.

There's a lot of different options but disk2vhd seems like the 'simplest' option since modern versions of windows support VHD natively. The image would take the space taken up by the contents of the drive and you can grab files off it at leisure. I don't think there's any compression option but the ease of mounting makes up for it.

Likewise, most decent backup software would let you mount a backup. I've used macrium reflect free for this, but acronis or any other common backup software that does images would let you do this too.