No, the traffic from your IOT devices would still be on the same LAN as traffic from your other devices, which is a horrible idea. Perhaps you should re-evaluate owning IOT devices until you can afford your own router.

No, you need them to either be on different VLANs or physical LANs. They way you described is neither of these.

Can you not access the same content through the provider’s website(s) on your laptop?

No it doesn’t.

Unplug your modem for 24 hours, when you plug it back in you will probably have a new IP.