Does client mode shut off ethernet port on Airport Express 1st gen?

I want my desktop computer to connect to the internet by ethernet in a room that doesn't have an ethernet jack. I am planning on buying an Airport Express 1st gen, and having it connect wireless in "client mode" to my Airport Extreme 2nd gen.
Will wireless "client mode" disable the ethernet port on the Express?

Follow up question:
Another option I'm considering is just buying a new Airport Extreme 5th gen and using that as my base station, and then use my old Airport Extreme 2nd gen in client mode.
1) Does the 2nd gen Extreme work in client mode (to access its usb port)?
2) Does client mode also shut off the gigabit ethernet ports of the 2nd gen Extreme as it does for the 1st gen Express?

I'm assuming then that the Airport Extreme 2nd gen does not have a client mode function with working ethernet ports... Is that accurate?