Drywallers messed up tile backer board, can I still tile

I would probably use unmodified thin set to cover the gaps and give a good base for applying the tile.

Could you do something a little decorative and use a border tile around the repaired area, sort of like a picture frame? Then use a different tile to fill in the area. The border would make the change in depth harder to see.

If there is moisture still there, then mold and mildew will return. If the problem was not fixed, then it will return. Bathrooms are particularly bad in this respect, since moisture is a natural consequence of using water. I would expect to need to redo things like caulking every few years to fix this. It will not last forever.

1. Scrape/pull off the old silicon.

2. Apply a solution of bleach to kill the mold. Be careful when you apply it, as bleach can damage the clothes you are wearing. Keep the area ventilated well.

3. Let it dry. If you are in a hurry, a fan blowing on the area will help. A hair dryer can also speed that process considerably.

4. Apply new caulk in a smooth bead. I use my finger to spread it along the joint. Definitely use a high quality silicon caulk with a mold/mildew inhibitor. These are caulks that are designated for use in bathrooms.

5. Allow it to fully dry before you get it wet again. Give it a day or so. It will skin over in an hour or so, but make sure it has properly cured before use of the shower.

Expect to redo this process in some number of years. The exact time it will last depends on how well the area is ventilated.

Of course, there may be mold in the wall itself. This is difficult to deal with unless you tear the wall down to replace it completely, and even then water tends to attract the stuff. Mold spores are everywhere.