As I stated many times below you need to buy a book and start from page one. There is a ton of information you need to learn before even diving into making a network. If you want to do anything network based you will need to know all what a network entails.
What the difference of a switch/hub/router/bridge are
and much, much, much more.
1. You do not HAVE to set up your server to be a DHCP server. You can have the router do that function. As for the DNS feature on your router, you can turn it off once you have set up your server to be a DC (domain controller).
2. Turn off your DHCP and DNS functions on your router.
3. Write down all of your settings if you are afraid you will mess something up. If you mess with the DNS/DHCP settings and screw it up you can restore it to factory defaults and BAM you are good to go.
4. You need to get a book and study. You can't expect to just be able to blindly do this. There is a bunch of configuration that goes into this.
To set up a domain you will need to run dcpromo from the run command. Once you have your server configured to be a PDC (Primary Domain Controller) then you will need to set up DNS and DHCP. You will need to set up DNS forward and reverse look up zones for your network. You will need to set up scopes for your DHCP server.
You will also need to make sure that you are using correct networking addresses that fall within the same subnet.
5. Much more complicated than what you are doing. But here is a diagram.
6. Very possibly yes. You may not be able to use a different modem other than the one they provided you. Chances are this will be the case.
7. You should get a book and study. There is a lot of stuff you need to know prior to even trying to set a network up. I highly recommend a set of books like THIS