1. Sounds like a bug, the software should re-apply any custom set names on startup but it sounds like it's overruled, I'll put it on my buglist to investigate & fix.
2. I've looked at BVH support in the past unfortunately this 25+ year old is not a very good fit.
- many applications discard any positional data (except on the root joint)
- it was designed for (human) hierarchies of bones and can't save things like custom attributes (for buttons for example)
- it doesn't handle things like geometry
- it saves in Euler angles and can suffer from gimbal locking issues
- it doesn't save timestamps per object so can't properly support variable framerates and/or framerate fluctuations very well