Hi Heath, welcome to Vatsim!
I'll briefly answer what I can for you, but your best bet for information is going to be reading through the Pilot Resource Center located here:
https://www.vatsim.net/pilot-resource-centre1. There will be multiple servers to connect to, select the one closest to you. It sounds like yours is going to be the USA-E server. Once you do you will see a list containing the available Air Traffic Controllers, if any are online at the time. You'll need to figure out which one to contact, and how that all works is located in the PRC I linked above. If nobody is online, you will need to tune 122.80 and state your intentions on text.
2. Nope, you can use any aircraft you want to use.
3. I don't use X-Plane, so I do not know the answer to this. You can probably figure out which one it is by looking at your settings.
4. The callsign you use will either be a tail number, for example N123AB, or the airline ICAO code and flight number you are flying. There are very unique ones out there for military operations, but that is outside the basics. You enter your callsign when you are connecting to the network. Your callsign should not exceed 7 characters. So it could be N123456, or SWA1234, just do not exceed 7 characters.
Go ahead and read through the PRC I linked above, that should answer most questions you have. If you still have questions feel free to ask here and somebody will help you out!
Another way to begin learning how to properly use and fly on Vatsim would be to join an ATO and go through their pilot certification process. They will teach you the basics of Vatsim, flying the pattern, using instruments, etc. Most of these are virtual airlines, but a few like
VATSTAR are dedicate to pilot training only.
https://ptd.vatsim.net/atosHappy flying!