First question I have, if Verizon Android is a cell phone, how can it connect to the router via Ethernet?

I think it is possible to understand whether these three (actually four) connection solutions can meet your requirements through field testing.

If it is an MT3000, it does not support customize change the antenna, and its WiFi antenna is built-in.

Since the router has its own system and better antenna, I think it will theoretically be more stable to connect to public WiFi through repeater, but I am not sure whether the switch link response time of failover can meet your requirements.

Probably yes.
There may be some problems with one phone, and the other phone may continue to work.If multiple links are connected to the router, all mobile phones will only be connected to the router. When the router's superior network is unstable, all mobile phones will be stuck.