I am replacing an entire bath floor system and plan to install 3/4 plywood with cement backerboard for ceramic tile.
Question: Do you install a vapor barrier between the plywood and the cement board? Also, should the seams around the backerboard be sealed with joint tape, etc. Also, what kind of screws to use for both subfloor and backerboard. Also, should adhesive be applied between plywood and vapor barrier/backerboard?

No vapor barrier needed here. I install Hardibacker with their special screws and modified thinset. The same thinset is used to mud the seams with fiber glass tape and then used again to set the tile. Hardibacker comes in both 1/4" and 1/2". 1/2" normally for walls but in your case I'd use it for the floor too.

When installing the new subfloor, you want to use either tongue and groove subfloor plywood or you can use square edged exterior glued plywood rated because or better. Anything rated as sheathing or "CD" had too many voids to offer the correct support for your tile. If you choose to use square edged plywood, you will need solid blocking between the joists to support the panel edges as they span the joist bays. You also need to use subfloor adhesive on the joists and fasteners are installed every 6" where panel edges end on a joist and every 12" where panels just cross joists. Now those are minimum guidelines and more fasteners with closer spacing ensures a floor that is tighter to the joists and heavier compression of the adhesive beads. Now even if it's more convenient to run the long edge of the plywood along the joists, you do need to run it across the joists as that is the strength axis of the plywood.

After that, you can install any 1/4" cement board using thinset and cement board screws. If you need a higher floor, then you could use the half inch cement board. Either way, the floor will be the same strength. If you want a stronger floor, then 3/8" or thicker because rated plywood underlayment screwed down to the subfloor, with the long edge also crossing the joists (do not turn the plywood 90 degrees for tile installations) and then you can use a tile membrane such as Ditra or Noble CIS. Ditra and CIS are 1/8" and 1/16" installed, respectively and are superior to any cement board installation.

In asking about the vapor barrier, if you desire waterproofing of the floor, either of those membranes can be used as part of a waterproofing system or any of the cement boards can be covered with any one of several trowelable membranes on the market.

In every case, the thinset you use must always come from a bag. If it comes ready mixed in a bucket, it is not thinset and will only lead to problems later.