I'm quite sure my Windows PC is infected beyond salvation and I want to format it and reinstall everything. However, I have a 128 GB SSD where I keep only the OS and a 1 TB hard disk where I have programs and all of my files. If I boot from Linux to save whatever I can save, what should I save and what should I be careful about completely deleting?
If you have backups from before you got infected that could not be accessed by this system after it became infected, just nuke everything and restore from those. If not, continue reading.
I guess all executables are better deleted as a malware with control of the system could've replaced them with malicious versions? And what about other kind of files?
Office documents and PDF's are the big ones to make sure and remove. Both of them are pretty actively used attack vectors, and just loading an infected one is quite often enough to compromise the system.
Others include (but are not limited to) JAR files (which are Java executables), any virtual machines you might have on the system (they're probably infected), and anything inside
In particular I'm concerned about my emulators and ROMs of retro consoles. I'll probably delete all the emulators, but are ROMs at risk? Can they easily be corrupted to become malicious software?
The ROM's are probably safe. While they actually are executable code, they aren't native code and they aren't highly portable, so they're not easy or attractive targets for malware. Unless you can reliably determine that you were infected through one of them, you can probably safely assume they are not infected.
And in general, can I keep documents, videos and music?
Documents other than plain text files should probably be wiped. Videos and music may or may not be safe to copy over, they're not hugely common attack vectors (so the same argument as used for ROM's (if you can't determine that you were infected because of them, they're probably safe).
And if any executable slips through and I don't manually execute it, could it cause any damage?
If you're doing this right, no executable files should remain from the old system, at all. This is not a case where you should be using the built-in factory reset provided by Windows, you should be wiping the drive and doing a clean install from installation media (which should be prepared on a separate system if you need to create it). In an ideal scenario, you should actually get a completely new system (or, at least, a new motherboard, CPU, RAM, hard drives/SSD's, and any add-in cards), not just reinstall, because you can't be sure the firmware isn't compromised.
Also, when copying out old data, make sure you copy just the file contents and filename. If you copy things in a way that preserves any further metadata, you run the risk of accidentally copying contaminated data (malware likes to hide in NTFS alternate data streams).