how does a usb wireless network adapter read or connect to the ethernet port in my wireless gateway?

Answer 1

same frequencies as cordless phones.
The router detects the wireless signal
Determines the device by its mac number ( id number from manufacturer)
With the dhcp hands it an ip.
I am not sure when the ip is handed over - at initiation or only if the device connects - i.e device is connected either by user or setting on the computer os and if the encryption is correct is so enabled.
If you go into the status - lan of your router you will see the list of network devices ( can be wireless printers etc) that the router has given an ip to - sorted by mac address
If you turn on the mac filter in the wireless you can scan and see which network cards ( mac addresses) are currently on the system ( as wireless)

Answer 2

your gateway connects to the cable, or phone line )for dsl or highspeed) internet.

the ethernet port for your gateway should be labelled:
LAN (to connect a ethernet cable to your computer)
WAN - to your ISP's cable, or phone connection

The usb wireless network adapter connects directly to your laptop or desktop. It senses a wireles network and will try to connect. ( provided you have the drivers installed for the usb network adapter)

Answer 3

It connect via wireless, you do not plug into the router at all.