Firstly, thank you for providing so much information.
I suggest that you approach this problem in 2 stages:
1. establish the internet connection.
2. establish the wireless connection.
To establish the internet connection, set up the gateway at the phone point through a microfilter and connect your pc to the gateway by ethernet cable. Log into the gateway and put your ISP username and password in the broadband set up page. Save the settings if necessary (some routers save automatically).
Check the gateway's network IP address (it will be something like 192.168.1.1) and set the DHCP server to issue addresses in the range 192.168.1.3 - 192.168.1.9 (the first 3 sets of numbers should be the same as the first 3 sets of the gateway's IP address).
On the pc, go into the TCP/IP properties of the network connection and tell it to obtain an IP address and a DNS address automatically.
Open a cmd window (Start - Run, type cmd [Enter]) and type ipconfig /release [Enter], then ipconfig /renew [Enter].
Now check to see if you have an internet connection. If you have, set your gateway's firewall and your pc's firewall. Make sure that the pc's firewall will allow the IP address range you set in the gateway.
Once that is working, you can set up the wireless network. I suggest that you do this close to the gateway so that you can reconnect by ethernet if something goes wrong.
Make sure that the wireless part of the gateway is switched on in the setup pages and set the SSID (the wireless network name) to something you will recognise. Disable any encryption.
Disconnect the ethernet cable, install the wireless drivers and attach the wireless adapter to the pc. Again, set the TCP/IP properties of the wireless adapter to obtain an IP address and DNS server address automatically. The adapter software should show the wireless networks that it has detected by SSID. Highlight yours and click Connect. After a short while the adapter should log on to the gateway and obtain an IP address. You should now be able to access the internet wirelessly.
Now you can consider protecting the wireless network, but get this far first.