Don’t put up drywall under the backerboard. I am a believer in the felt behind the cement backerboard practice, but there are a lot of professionals that question the need for it & install without it. Your tub should fit snugly to the studs & have a support ledger under the long side. You should fur out the studs so that the backer board comes down over the lip of the tub leaving a space between the bottom of the backerboard & the top of the tub to prevent moisture wicking of the backerboard. Also make sure you tape the backerboard corners & joints with fiberglass sheetrock tape & mud them with thinset & you should be ready to start tiling.