Short (as possible, in this case) answer...
1) Depending on your exact configuration... Possibly. More on this in a second.
2) Technically, yes. A router will have two addresses.
3) Yes, yes, no.
A router handles traffic traveling between two networks. In this case, between "the internet" (we'll just leave it at that for now) and your own network (home, work, whatever the case may be). Part of this process is what is called "Network Address Translation (NAT)". Your router has a publicly known IP Address. This is the address anyone else on the internet will use to communicate with any device on your network. This address is unique, not used by anyone else in the world (basically). Typically, you will only have one public IP, no matter how many computers you have plugged into the router. The router will keep track of all of your "private" IP Addresses (these are the addresses used by your computer(s). These addresses are not unique in the world, and can be (and are) used on any other private network in the world. However, they are unique within your own private network.), and which traffic from the internet is destined for which private address. The router then "translates" the private addresses into your public address. This is how you can have multiple computers connected to the internet through one router and one public address.
The basic explanation of the "Default Gateway" is... The computers on your network don't know how to communicate with anything outside your own private network. So, any time they need to get "outside", they simply send the traffic to the Default Gateway, which has the responsibility of sending the traffic on its way.
So, the most correct answer to your final set of questions is... The "public" address of your router would be what a site uses to ban you. This has nothing to do with your Default Gateway. Your public address is controlled by your ISP.
And no, I won't tell you how to try to sneak past a ban. We've already had a thread on that this week.