Should I build the tub deck first and incorporate its framing into the
framing of the room, or should I drywall first and build the deck
after?

Your choice. Either way will work fine.


I plan on useing Kerdi waterproofing around the tub. Should I use
drywall or cement backer board to support the tiles on the walls?

cement backer board


The weight of the tub is to be supported by the floor and not the tub
deck.At the same time I want a close fit between the tub and the tub
deck so I can waterproof the joint.By the time I add cement
board,Kerdi liner, mortar, and tile to the top of the tub deck, I
don't know how thick this will all be.Its hard for me to accurately
build the framing for the tub deck.Maybe I can build it slightly
higher and use extra mortar to build it up to the tub edge?

Build it slightly higher. Pour a bucket of mortar onto the floor, set the
tub in place and press it into the mortar.