Indeed, your concern is quite valid, but don't worry too much - what's done is done. Once the RAM is replaced and good again, the computer carries on; replacing the RAM is the only precaution you need to take. What you do with your data files, however, is an entirely different matter.
Question: what should I do once new healthy RAM has been installed? Reinstall Windows (I'm using Windows 7, 64 bit)? Is there a risk that I will be presented with nasty surprises in the future if I don't?
So long as the operating system is still functioning correctly and not crashing, it's unlikely that anything on the disk was corrupted. Due to how you found the bug, it's possible (ASLR really makes these types of thing much more apparent) that you only encountered it when using that part of the memory, and furthermore, Windows never modifies important system files during normal operation (i.e. not installing updates to the OS).
I don't think you need to reinstall your OS, but I would avoid using the computer until your new RAM arrives.
What about personal files? I have backups of some of the files but for newer files I'm not sure I can even trust the backups. It's going to take me many hard hours to manually replace files with older versions, or compare checksums.
Really, the only files that could be corrupted are those that were being edited in memory, and then entirely written back to disk. If you changed the files, checksums won't do you any good as you don't know if the files are good nor bad - and yes, you would have to manually check in these cases.
Remember though, the files won't just be corrupted by themselves - you will have to have edited the file, and re-written it. Fortunately, photographs that are corrupted by a bit or two are usually recovered pretty easily, and even if visual artifacts did persist, you can touch them up manually. The hard part is going to be determining if a file was corrupted, and what to replace it with - and that's a question only you can answer.