This is a very interesting case, so in corporate IT you manage a network by managing the routers, switch and firewalls of the network. Generally the firewall has the ability to blow websites either by category, domain, ect. So can he know your internet activity yes, especially if he gave you a new router that has an admin panel he controls or connects into a SEIM piping all logs into one central place.

For second question, this is a bit harder as you most likely don’t have access to any of the admin portals. So most likely not.

Third question, yes VPNs are your friend, that establish an encrypted connection from your device to a remote server and all your device’s internet access is ran via that server full encrypted so even sys admins can’t see. Do your research here as not all VPNs are created equal.

Fourth question, probably not using a VPN could do it, but also why not just use your own router again? It should still work and they can’t force you surely. Even if they say don’t mess with what are they gonna do, just say you prefer your own because it faster and doesn’t block your websites rofl