If I get a small screen (15") for the server, does that mean when my clients/users login via team viewer the will only be able to view at the resolution my 15" puts out? Instead of their 24"+ screens?
That's because Team Viewer is a remote desktop simulator. If you are doing this proper you should make a proper domain login so the VMs on your server act as a cloud machine rather than remote desktoping in.
You'll have to make a thread so people more in the know about that kind of stuff can help you out. I know of it, know it's possible, and have done it before but haven't completely set it up so couldn't tell you step by step how to do it.
I do know that you can have a small monitor on the server rig and RDT with a bigger res. I do it here. Could just be the program at fault or they need to change their res settings while in the VM. Most of the time my servers don't even have a monitor or even a GPU.