Laundry Room Remodel Using Existing Vent

I would like to re-model our existing laundry room (see attached drawing) with a clothes washer and sink, using the existing 2” vent if possible, so that I don’t have to cut additional holes in the roof. But I have several questions:
1. Do I need to back vent at all (dashed lines in drawing) or can I simply run horizontally to the existing vent?
2. Assuming that I do need the additional 1½” back venting, are the horizontal distances to the 2” vent too long?
3. If the basic plan is ok, do I use san-T or combos where the 1½” vent lines intersect the horizontal drain lines?
4. Washer drain must be 2”, but for the sink drain, can I use 1½” from the trap to the vertical? I have seen contradictory info on that.
Any corrections or suggestions for a better way of doing it would be most appreciated.