Modem/Routher

You will have to configure it in AP mode, if it has that mode or disable DHCP, firewall

You dont need another modem.

What you should understand, that device supplied by your ISP(Virgin) is All-In-One - "modem/router/switch".

You only need one modem - you do

You should not have two routers, unless you know what you're doing.

All you need to extend your network. This can be achieved by various solutions, powerline adapters one of them. There is pros/cons for every solution you consider.

Ethernet cable would be best speed-wise but require cable run if you dont have one in place - are your rooms wired with Ethernet ???. Is there phone or RJ45 socket in that particular room?

Powerline is near plug-and-play, however, most likely will limit the speed at destination location

Other potential network extension option would be WiFi Mesh

Would you be able to run Ethernet cable from router to that room? I mean in suitable aesthetic manner, inside or even outside the building.

If not, then perhaps powerline or mesh is better for you.

and no, i dont know what you mean by: "I don't think I'm allowed mention whatever i want if for but I think u can guess?", all i can think is - you need to extend your network