I am looking for a way that would allow me to have one server run all
Windows (7) installations, while the users get basically a monitor,
mouse and keyboard.
Does such a solution exist?
Yes, such a solution exists, but you won't want it. It will be enormously expensive and introduce a single point of failure for a small office. Your scenario is nowhere near the use case scenario for VDI and barely that of Terminal Services. VDI by itself won't solve the whole issue of managing different systems either. In essence, you'd spend a ton of money to have one choke point and only about a quarter of your problem solved.
My situation is as follows: I'm in a company with 12~15 employees. All
of them are currently using a separate pc with a Windows XP
installation. Managing these systems is very time consuming and
inefficient, and keeping these systems up to date is virtually
There are places that concentrate 200+ users per administrator. If you are having so much trouble, you either aren't using Active Directory correctly or at all. A single SBS instance will handle this situation with enough potential for automation and management of your scant few PCs such that you will hardly know what to do with your spare time.