apple remote desktop, using remote computer independently

I'm not clear on whether the "home" computer will be the remote system, but ARD just allows you to take control of a remote computer (along with other update and management functionality), so any program you wish to run on that remote computer must be installed on it. If by "home" computer you mean that this system would be the one from which you'd be controlling the remote system, then no, the software does not have to be installed on that "home"/"adminstrator" system.

Regards.

No. Again, what ARD gives you is the ability to take control of the remote computer. It does not give the administrator /home computer itself any of the software nor capabilites of the remote computer. And unless the remote computer is on the same network as your home computer, in which case there are other ways to share that system's Internet connection, you probably couldn't connect to the remote system at all from a system that lacks Internet capability.

Regards.

Since you're taking control of that remote computer, you can run any software that's on it just as you would do if you were sitting at that remote computer.