The handle lock on sliding glass door ( classic by Mayfair)

Hello! Your exact handle set is no longer made, but we have a few that could be used as substitutes. Look at our 82-033 to start out, which is probably the closest to your old design. This doesn't come with a key lock, so you'll also need an 82-310.

Finally, based on the problem you described, you'll definitely need a new mortise lock. It would be either our 82-211 or 82-214, depending on size. Honestly, reading your post more carefully, there's a very good chance that your mortise lock is the only thing that has to be replaced. You could get away with keeping the same handle set. It's worth experimenting with; a good way to confirm is to remove the mortise and try to throw it manually. If you run into the same problem with the falling hook, then the mortise is the culprit.

Either way, look these parts over and let me know what you think.

Yes, that should be totally fine. The 82-306 outside handle is fully compatible with the inside handle that comes with the  82-033 set.