I am interested in developing an app for my new iPad purely for my own use. (well to start with at least)
Is there a way to develop an app without a mac?
Officially, no. Realistically, unless you like wasting countless hours, no.
Can I install the app only on my own iPad without having to sign up to the right to publish it when I might not even want to do that?
No, you must be a paid developer in order to push to anything but the simulator.