I usually do it through my laptop. You can USB-tether your Android's Wifi connection to your laptop and then make your laptop as a hotspot so that people can connect to it. BTW, I tested it on Linux and it worked perfectly.

It is possible with software like Quagga - essentially it turns your device into a router, which, I would assume, could include smartphones.

I know smartphones have some sort of setting that allows other devices to connect to it..mobile hotspot, but I'm not sure if it's possible with a dedicated router

I mean it would work, but it would be a PITA to set up and I wonder what kind of performance you would get, my guess is bad

If your smartphone can function as a router then why would you need another router?

That is basically possible though you need a routing software on that phone to route the internet traffic to and from that external router. Phones usually don't have that built in but maybe you can program that or find a suitable 3rd party software.

No idea