I downloaded a JB 4.2.2 CyanogenMod ROM from the original development's thread. I tried a couple of similar ROMs previously, but my home button doesn't work correctly on apparently any Ice Cream Sandwich + ROM. Essentially what happens is, when the screen is on, the home button is completely useless. Pressing it down, waiting for a few seconds - it just doesn't respond to any command. There are almost no options in the phone related to hardware buttons, and if I, for example, select an option to make holding down the hme button bring up the multitasking window, it still doesn't do anything. The only situation where I observed it doing anything is when the phone's screen is off - pressing the button instantly wakes the device up. I tried flashing the badass kernel, though it didn't really do much. Other hardware buttons seem to be working just fine. I had no such problem on CyanogenMod 7.2 and when I restore a backup with this system, it still works correctly. I don't have problems with getting into recovery either.
Another thing with this ROM is, the lockscreen doesn't appear, no matter if I set a security pattern, set it to slide or whatever else. Pressing the wake button or home button instantly wakes the device up.
I'm not sure whether it's a ROM-specific problem or what, but regardless, as a new user, this is the only forum I can post this problem to. If anyone knows a solution or if such a problem was solved previously, I'd be happy to learn the fix.