Airport Extreme WiFi Bridge?

Can the airport extreme connect to the WiFi router in room 1 over wifi only and provide a ethernet connection to the NAS, Xbox and PC and supply them internet connection?

Yes, if the WiFi router in Room 1 is also an AirPort Extreme, or other Apple router.

No, if it is not.

But I have to have the NAS, Xbox and PC on gigabit lan for file transfers between them.

You will not have Gigabit speeds over the network since the connection will only be as fast as the wireless "hop" allows between the "main" and "remote" router. This will likely be something in the 100-240 Mbps range with a very good wireless connection.

This will not work this way.

If you put WiFi card into your PC you can share internet via Eth port from it.
Then use Airport Extreme as FastEthernet switch only (what a waste) :-)

So any data from the PC will be transmitted to the router in Room1 over wifi then back over wifi to the NAS?

If you are copying data from one device to another with devices connected to the "remote" AirPort Extreme, the signal must go all the way back to the main router and then back out again to the "remote" router. Things will be ultra-slow since the signal must make two wireless "hops" along the way. With a really great 5 GHz wireless connection, you might see 150-200 Mpbs or so.

By the ooks of it all the devices that need to communicate over LAN are in room 2 so this is where the router should be. This way you will get FastEthernet between all devices in Room 2.

All you need to do is to get the WiFi to be sharred on those devices.

Home made fix is as I mentioned by using PC to share it's internet from WiFi.

Very likely your WiFi is faster than your internet connection so no loss of speed.

This way you will get FastEthernet between all devices in Room 2.

Fast Ethernet is only 100 Mpbs....not very "Fast"

Maybe you were thinking of Gigabit...1,000 Mpbs?

Yes Bob I mean 1Gigabit

Bed time for me :-)

good night