Hello Jason and welcome to Veeam Community forums!
Should my Veeam server be joined to the domain?
No, you don't have to join Veeam Backup server to a Domain, but it is recommended.
Can I replicate my Veeam server (running it on a VM, Windows 7 64 bit)?
Yes, but please have a look at this topic for more information:
http://www.veeam.com/forums/viewtopic.p ... eam+server
I'm also licensed for vCenter, which I am installing in a VM. What backup methods can/should I use, virtual appliance?
You may use any backup mode if you meet the corresponding system requirements. Actually, the backup mode doesn't depend whether you use vCenter Server or not.
Regarding the previous question, can I, or should I run vCenter on the same box as Veeam Server?
I wouldn't recommend doing this in backup job performance considerations, if you want to backup vCenter Server then please have a look at the link below:
http://www.veeam.com/forums/viewtopic.p ... standalone
P.S. if you have any question regarding backup functionality, please look through our sticky F.A.Q. topic:
Hope it helps!