Yes, almost weekly, buut… do read One new, one broken, one beta, more updates | F-Droid - Free and Open Source Android App Repository
Basically the developer pushes stable versions when they want. If you want to use the latest betas go to the upper right menu, toggle Betas 
Yes, press “Submit feedback” and tell them “Play is buggy, it erroneously tries to update apps with the wrong signature”
known issue Google Play Store tries to update my F-Droid apps :-( 