It would have been so much better with a DSLR

LOL!

Attached is the tracert photo.
tanstaafl.
That tracert photo shows you are in facet connected to the internet correctly. You are resolving IP addresses into domain names, you are going from hop to hop... Problem is not in TCPIP, and it is not in radio association with the WAP.
I think the issue may be in firewall or some browser setting. I suggest you
1. Try to disable windows firewall, just for a test: control panel, windows firewall, disable. - edit: sorry, I read in your first post you already did so.
2. Try a different browser (not permanently, fo course), just to see if it is browser specific.
Edit:
3. Do messengers work? Do email clients (Outlook, or similar) work?