If I jailbreak my iPad, will I be able to use the App Store to download apps and also test the apps I develop?
As @Timothy pointed out in the comments, the App Store will work normally on a jailbroken device. As for your own apps, you can run them, but if you don't have a paid developer account it is harder to get them on the device. If you have a paid account, you can use the normal methods to install and run them.
A device which is not jailbroken will refuse to run any app which is not signed with an Apple-approved certificate. When you jailbreak, the code which performs the checks is modified. It will still only run signed apps, but those apps can be signed by any certificate. Therefore, you can create your own certificate to sign your apps with. However, the normal methods of installing an app will perform the check and refuse to install it. This means you need to find an alternate method of installation, most likely using SSH or USB to copy a package over and modifying permissions.