Ok, so these are what I'd do if I were you.
—Try your copy in a different PS3, or
—Try a different copy of GT5 in your PS3. If that doesn't work,
—Download the PS3 update. If that doesn't work,
—Delete your game data (after backing up your save data), install it. If that doesn't work,
—Backup all your data, format your hard drive, reinstall. If that doesn't work,
—Replace the hard drive, reinstall. If that doesn't work,
—Start saving for a new PS3, because you'll probably need it.
Also, keep backups of all your saves and other data as often as you can be bothered to. I didn't and lost quite a lot when my fat PS3 YLOD'd 5 hours into GT5 (on the release day, as well!), though to be fair I haven't played any other game for any significant period of time since GT5 so that wasn't a huge loss.
However, I too have had strange behaviour since 1.10, and I don't have PS3 OS 3.65. Things like you described (loading a race into a black screen, then quitting back to the GT Life screen), quitting back to the GT Life screen immediately after a race ends, loading the pre-race screen but not having a menu appear for a good couple of minutes... The track hasn't disappeared for me for a long time, though, it's been months since that last happened.
Sounds to me as though your data is corrupt, though, I wouldn't bet on it being the disc or your PS3 getting ready to die (the signs of that are more obvious than just crashes, things like the PS3 turning itself off happen long before that), it could be a corruption from the download not being 100% right (unlikely) or your hard drive just getting old or tired. I've had that happen to me twice with my old PS3, each time the drive was under two years old.