I don't know off hand if the Realtek WiFi needs proprietary drivers. I would tend to assume so, as it seems that most wireless adaptors do. Did you installs proprietary drivers for it? Given that it is working at all, I expect so.
Edit: wait on, just thought to look at the specs here. This machine has this Realtek adaptor
Not the same one as yours, but it is fair to assume that you need drivers for it.
description: Ethernet interface
product: RTL8111/8168B PCI Express Gigabit Ethernet controller
vendor: Realtek Semiconductor Co., Ltd.
The wired connection shouldn't need any drivers or anything else, for that matter. It should just work. I believe I have seen reference to that ethernet adaptor, maybe even on one of my machines. Any rate, I wouldn't expect to have any trouble with it normally.
I think trying another browser is not a bad idea. Opera would be the first candidate.
You could also try this one:
http://www.elinks.cz/ That should run even if you are booted into a command line only environment. The interesting thing about that would be to exclude some fault in the GUI.
A hardware fault as such is also possible. The thing is, you say you are seeing the same behaviour on both a cable and a wireless connection. You also say a different machine is connecting without a problem to the same router. That all seems to exclude most possibilities for a hardware fault. You should, of course, do the simple tests like swapping out the ethernet cable, plugging in to a different port on the router, and so on. I suspect you wont find anything, though.
Another thing to try is booting into the live environment to see how it behaves there. That is the "try ubuntu without installing" option on the cd or USB you installed from. Try at least the cable connection from there to see if it works properly or not. If so, you have a clear indicator that something is not right with your install.