No, it’s not government funded. It’s funded by the people from a yearly licence fee, so has no ads. This is how it stays politically impartial too (mostly).

F-droid store has a number of health tracking apps. I haven’t used any but saw them while looking for apps that would integrate with my Garmin watch.

I haven’t tried to set up music yet, but getting a lot of false lookups on tv episodes even after renaming to the generally agreed file naming format. Still playing with the setup so may get better. Linked it to my kodi on rpi4.