I am interested in developing an app for my new iPad purely for my own use. (well to start with at least)
You can develop for iOS without a Mac by using Adobe Flex. The Flex SDK is free and Open Source, and includes a compiler + packager. If you want an IDE, you can get Adobe FlashBuilder (not free, but with a free 90-day trial), which makes your development time much more productive. You would not develop in C, C++, or Objective C, but in a combination of ActionScript (a dialect of Javascript) and MXML (markup language, mainly for rich GUI layout).
On the minus side: you have less control over what you can do (you can still do a lot) and you cannot use the iOS SDK directly.
On the plus side: it is very easy to develop great looking apps, and with very little changes, you can recompile them for Android, Blackberry Playbook, Windows, and Mac OS X.
You do not technically need a Mac to develop an app, unless you are serious about it. There are couple of solutions available to you for developing mac-less.
First, there are some services that port your HTML5 web apps into an iOS app, so you only need to write your code HTML5.
Second, if you dont want to buy a Mac, you can instead buy Snow Leopard or Lion, and build your own "hackintosh" (a windows computer hacked to run off the mac operating system).
Finally, you cannot make your app available for purchase in the App Store unless you are a paid developer. However, you could publish online as an HTML5 web application, or you could publish your app in a third-party app developer market (however you could only intall that app if you jailbreak your iPad).
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?
xcode for windows.
http://ipodtoucher55.blogspot.com/2010/12/installing-ios-sdk-and-xcode-on-windows.html
Not sure how new this is, but Dragon Fire SDK is 100% Windows based (you do submit your code to them for compilation), but at no time are you required to own a Mac.
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.