This is called WAN failover (or policy based routing depending on your needs) and that’s of course the job of a router.
ISP’s gateways plugged into the same unmanaged network switch.
This will not work and will cause issues with your networks performance
Yes. Coax is very fickle. It’s really easy to cause interference or mess up the signal one way or another. Even the change in temperature over the seasons has a strong effect on signal strength.
Call the ISP to get it fixed. Or if you’re up for the challenge, buy a coax stripper, a coax crimper, and some outdoor rated type-F connectors and fix it yourself.