How this issue is related with OP's one?
OP issue is due HDMI's port in MacMini. He is using a wrong cable. And since it changed to a good cable he is fine. This is a well known issue.
Your's is related with disconnecting after waking, which is a a well known and documented issue. You will solve this issue preventing your MacBook Pro from sleeping using the Energy Saver panel. Apple is being plagged with display disconnection after waking since Catalina. The only fix known is to prevent your computer from sleeping.
BTW, your issue is software related. OP's is hardware related.
Regarding bad bluetooth connectivity, try to reset Bluetooth module. Head to Apple Help Website for instructions.