Post a picture of the back of your router. It will be easier.
As the question of wired or WiFi, both will work. WiFi tends to be a little slower and has a higher ping/delay/lag. If you are trying to be the best at fast pace FPS style games then WiFi isn't recommended. If you are just planning on casually gaming then despite WiFi not being the best, you will likely still have a very enjoyable experience.
As mentioned before you can use a WiFi analyzer app to check and make sure the people around you don't have their WiFi on the same channel as yours, but the reality is you would likely need the ISP to correct that if it is a problem as they tend to not allow you to fiddle with the settings on their routers.
It sounds like you'll probably need a tech to come and look at your configuration options. It is possible that you could unplug everything and put it in a different room, but like @Urworstnit3m3r said, sometimes techs will only plug in what they need or they'll unplug everything else for one reason or another. You could probably figure it out on your own, but it could take a good bit of dedication, troubleshooting, cable chasing, and deduction work.
Tomorrow when I have a moment I'm going to try and connect the modem in said bedroom with each cable hooked up where it goes and see if the modem works and the laptop connects via wifi. I think this will be a good place to start.