I am interested in developing an app for my new iPad purely for my own use. (well to start with at least)
Yes, you can develop without apple computer (using hackintosh).
And yes, you can install the developed app on your iPad without having to pay for developer program. There are a tons of guides over the internet about running (thus installing) apps on ios devices without developer program.
check this: How can I deploy an iPhone application from Xcode to a real iPhone device?