I can't help with the hardware setup for the server, but I can help with the domain name registration questions.
If the hosting company that you currently have has registered the domain names under their ownership (many do), you MAY have to buy the domain names from them. Many of these outfits offering cheap domain registration give you a break on the domain name registration because they retain ownership of them IF the hosting account is transfered to another host, which is now going to be you. And, if they retain ownership, they can sell the USE of the names to someone else if you move the hosting account.
What you can do though is to go to NetworkSolutions.com, open your own account and TRY to transfer registration of the domain names to your newly created account. This may take several days to complete, so don't get into a big hurry. Network Solutions will contact the current owner of the names and ask permission to take control. Your current hosting company will have to answer, yes or no.
If the hosting company has not set the domain names to "private" -or- "no transfer", then Network Solutions can help to "grab" control of the ownership. Don't be afraid to ask them for help. Once you are designated as the "owner" of the domain name, you can change your own DNS settings from within your Network Solutions control panel so that the names point to your newly created server's public IP address.
Once again, this all depends on who has been set as the owner of the domain names. If you are designated, it will go quite quickly. But, usually it's all a very long and complicated process. The main thing to remember is that you need to take control of ownership of the names.