Can I or Should I Sand Drywall?
Sanding is not a good idea as the material you are trying to sand off is known as thinset or mastic ; it is a mortar like cement by the time you sand it off you will most likely damage the dry wall.

The process is to first scrape it with a scraper and then use a razor scraper - very time consuming and it will not be perfect. You can then come back with high grain sand paper if need be. Patience is your friend here..

If a bit of the dry wall paper comes off :


If the new backsplash will be covering it: Just use your
thinset/mastic liek normal it will cover it and protect it.
If it is in an area where the new back splash will not cover - apply a very thin layer of joint compound in multiple steps with a 6" - 8" wide putty knife.. allow to dry between steps. Lightly sand and you will be fine. 


Because your back splash is about the location of the divide between upper and lower wall - replacing the entire dry wall would require both upper and lower drywall sections (for that kitchen about 10-12 boards $150.00) - not hard and not very expensive .. just depends on how your removal work turns out.

Again this will be dependent on you as to if you want or need to replace / repair a section.