Will buying a digital full game from the xbox live marketplace still keep the data on the hard drive?

You can download it digitally and have your save be okay. The Xbox stores the save and the game installation as two separate pieces of data. As long as you leave your save alone, you will be fine. If you're really paranoid, copy it to the cloud before downloading the game.

Does your Xbox recognise the disc is Payday 2? If it does, you can simply install the game to HDD from a good copy of the disc (either from a friend, or rent the game).

This feature only needs the Xbox to recognise you have the game in the drive, once it has done that it will happily run the game from the HDD instead.

I used this method when I accidentally bumped my Xbox while playing Gears of War. Obviously if your disc is beyond all hope then you'll need to re-buy the game, but this method is a lot cheaper and definitely worth the try.

Yes, if you play a game on disc, and then start playing it on demand (as in, downloaded from the marketplace), it will use the same save file. It worked for me with Fable 3 and the original Crackdown; it will work.