MacBook Pro runs fans at full speed constantly

It probably wouldn't help, but you could try changing the keys directly in ~/Library/Preferences/.GlobalPreferences.plist or ~/Library/Preferences/ByHost/.GlobalPreferences.*.plist.

defaults write -g KeyRepeat -int 2
defaults write -g InitialKeyRepeat -int 15
defaults -currentHost write -g KeyRepeat -int 2
defaults -currentHost write -g InitialKeyRepeat -int 15

2 and 15 are the fastest settings shown in System Preferences. The changes aren't applied until you log out and back in.

The file in the ByHost folder doesn't normally have keys for the settings, but it has precedence over the normal .GlobalPreferences.plist.

You can remove the keys with the delete subcommand:

defaults delete -g KeyRepeat
defaults delete -g InitialKeyRepeat
defaults -currentHost delete -g KeyRepeat
defaults -currentHost delete -g InitialKeyRepeat

You tried swapping the battery, which means your system is a few years old... I have a 2007 MBP and had some similar issues here and there when my battery aged. Part of the problem was actually my power cord. It was flaky. If your cord is underpowered (e.g. using a 45 W rather than an 85 W for a MacBook Pro) you won't have enough juice to charge a dead battery, so it restarts as if one was not installed or won't restart at all. If your cord is flaking out, it may not be charging consistently (I have a spare cord that sometimes works, and other times needs to have the extension cord removed and re-seated before the MBP recognizes it and kicks on the charge indicator light in the MagSafe connector). Possible that your battery drain is faster than your power cord can replenish at this point. The fact that a different battery gave the same results again makes me suspect the power cord. I'd try a new cord (if possible) and see if that helps at all.