Hi. Welcome to Just Answer.
The drywall can stay. You will need to apply some backer board though. They have backer board that is only about 1/4" thick so you wont be taking up very much room at all.
The reason you need backar board is because the tile has to be attached to something that is going to expand and contract at the same rate. The backer board is pretty much just concrete sandwiched between figerglass matts so it has the same expansion properties that the thin set and tile have.
If you were to apply the tile directly to the drywall you would have most of it on the floor, grout cracked, or separated from the drywall, before winter sets in.
Something that I always suggest is the use of "acrylic add mix". It makes the thin set very strong and it wont powder up if laid a bit thick. Believe me it takes a big hammer to get it off!
No sprecial morter is needed. Just plain ole thin set and some acrylic add mix. Use the add mix for 1/2 of the water you use to mix the thin set with. 50/50 water/add mix.
Personal opinion...Large tiles on the floor are great. On the walls it just doesn't look right, and matching the lines coming off the floor is a serious pain in the back side. Use 6" or 8" tiles instead.
Please let me know if you have any other immediate concerns.
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