How to save my app files and/or data from my iPad 1 whose screen won't turn on?

• Connect the device to iTunes and force a backup using the menu command.
• Restore that backup to another device and carry on with getting the files off.

You should be able to borrow someone's iPad for little expense (a few beers or buy them a Pizza) and then wipe your data once you're convinced you've exported all the data you care onto the computer or wherever it's going to live.

Either go and see a genius to see if they have some magic they can use, like special hardware, or jailbreak it using a tethered jail break and see if that lets you SSH into it or something. SSH includes SFTP, so you'll be able to connect to it and explore its contents with most standard FTP clients.