I'm not a professional and it depends on the size of the area to be repaired, but it's pretty easy to cut out damaged dry wall and then mud the patch seams. You want to do whatever will maximize the clean, level surface.

And, yes, you will need to feather and sand your mud until it is dead level.

I wouldn't paint the mudded, repaired drywall before tiling, at least, I never have. And I've never primed drywall before mudding it.

If you've only read or watched videos, be sure you practice. If your repair area is large, even if just 'paper', maybe hire somebody?

I had small damaged areas where a 4" splash was removed. I spackeled the smaller areas and used a drywall repair kit for a hole left by the electrician. Then I went over any torn drywall with Zinsser Gardz which will seal the ripped area.

Thanks for the tip on Gardz, it sounds like just the product we need to seal certain areas. We will definitely practice before we go ahead on our actual backsplash.