Orbi compatibility with other brand access point

You probably know this but your use case does not need Orbi.  Orbi would still work but the satellite would use wireless backhaul to the router (at least until Netgear adds ethernet backhaul support to the satellites)  No reason to necessarily place the satelite in the cabinet in the family room.

You could 3 cheaper APs (like the Ubiquti) install a $50 ERX router in the basement to connet to the modem, install a $30 switch and connect it into the drop in cabinet in the family room and connect an AP and your AV equipment to that switch.    Cost wise it may be a wash unless you already have the extra router, switch, APs.

If you were to mix Orbi and Ubiquiti APs and use the same SSID and credentials - they should work but I bet you may run into handoff issues unless you have power adjustments just right.  You may have issues with interference.   Ubiquiti controller SW will let you adjust the RSSI for each AP and Ubiquiti supports DFS channels that may help with this. 

I should add...  I had two Ubiquiti AC APs and replaced them with the Orbi because the coverage provided by the Orbi was significantly greater than the coverage with the two Ubiquiti APs.  i.e. You may find that you don't need any other APs but the Orbi.  You need to find out what works for you.

Good Luck!