I'll do my best to answer your questions, but you can read this for all the boring details: http://en.wikipedia.org/wiki/Proxy_server
1. I'm fairly certain you can only have proxy options for a browser, not a router...but I haven't looked at my router's settings in a while, and I don't have access to it right now, so I'm not sure on this one.
2. Yes, using a proxy changes your IP address.
3. I understand the question, but I'm not positive of the answer, as I do not have experience with proxies.
Basically, the premise of a proxy is this: you want to change your IP address for some reason (usually due to restrictions of your country of residence). Using a proxy reroutes your connection to all websites through the proxy server
you chose to use. This makes your IP address appear to come from the location of the proxy server instead of your actual location. Think about it like tracing a phone call. If you put up some sort of filter/firewall, you can give the call tracer a false location (I assume this is possible, I don't really know...it just seemed like a good analogy).
A good example is this: I wanted to watch a certain video, but the website would not allow me to because I don't live in Canada (wasn't my location). If I had used a proxy that routed through Canada, I would have been able to watch the video. But I didn't care enough to go to that trouble.
If you want to experiment with proxies, I have no recommendations, as I have never used one myself. Just be careful if you have to install software, sign up on websites, etc...
You can use http://www.ipchicken.com/
to see how your IP address changes when you start messing with stuff.
I hope that answered your questions.