OK, I'd look for a Mini that can hold at least8 GB of RAM...
But I don't like that one because the Shared Video, I'd opt for...
A real Video card & 16 GB of RAM possible.
An older even PowerPC Mac can do all your requirements. However bare in mind the following.
Mac OS X 10.4.x cannot run the latest iTunes, it is however possible to install Mac OS X 10.5.x (Leopard) on a PowerPC Mac mini which currently can still run the latest iTunes.
USB hard drives are slow. FireWire would be better.
The ordinary version of Mac OS X (any version) will not act as a Time Machine Server for other Macs to back-up to. The Server version of Mac OS X 10.6 or later can do this which means you would need an Intel model Mac mini.
I have setup an Intel Mac mini running 10.6.8 with iTunes and Logitech Media Server to do pretty much exactly what your are trying to do. It works fine.