Will 13 run on Windows 2003 Server?
- Specs for new server- on those on Filemaker's site truly sufficient?
- Is anyone using Apple Mini with upgraded ram and OS10 with good success?
Sounds like you partially answered your own question. The technical specifications page provides information on both the operating systems that are supported as well as hardware requirements.
As you can see, Windows Server 2003 is not supported with FileMaker Server 13.
As for a Mac Mini - your mileage may vary but a Mac Mini can be a great option for FileMaker Server 13 in a single machine deployment. I'd be a little less excited about suggesting a Mac Mini be used as a web server in a two machine deployment but it would likely be OK if the number of concurrent FileMaker WebDirect users was kept on the low side. But again, it is really tough to say if ANY machine is up for a task since so many things go into recommending hardware - like the number of users, the design of your solution, etc. We have some information about hardware consideration in our knowledge base here...