Has anyone else had this problem (in Thief Gold, at least, if not in the others as well)? You shoot a rope arrow into an object over your head but the instant you try to jump on the rope, Garrett lets go. Anyone have any insight on this problem?

Aren't the rope too close to a wall or some object? I recall something like this happening in such sutiations. Try shooting the rope arrow from a different direction, or into some other part of the object.

No, as I recall, I shot a rope arrow into the ceiling in the center of the room but Garrett still immediately released upon touching the rope.

That's a well known Thief Gold bug unfortunately. Sometimes if you keep trying to jump on you'll stay and other times, you have to shoot another rope arrow in and climb it. It's good to have multiple saves so you can reload and try again if necessary. It's not something that happens often in T2.

This happens in the Bonehoard level sometimes. My advice is to create a separate save before you jump into the pit trap with the face with the gem eyes.