privacy – Are my devices and information vulnerable in a public network? what are some solutions?

Get a VPN, such as Mullvad, to encrypt your connections to public networks.

is my private information (Banking info, private photos, etc) in danger?

If they don’t use TLS (HTTPS), then yes. An attacker could view or modify any data.

Could someone spy on me through my webcam?

Not passively. They’d have to manage to get you to execute malicious software on your computer (either by getting you to download and run it, or by tampering with another downloading executable that you’ll run), or you stream video from your webcam without using TLS.

I found that youtube was streaming to a SMART TV that was connected to the same network without me even allowing it, it just automatically did it.

That has nothing to do with public Wi-Fi. Check the smart TV’s manual, maybe?

If I buy a router and connect it to this public network, would my devices be safe then?

Not if the router has to connect to the public network over Wi-Fi.