Q.1 answer: UPC lists the total fixture units you describe as 10.5 f.u. and shows that you may have 16 f.u. on Vertical drainage piping and 8 f.u. on horizontal. So you can run the washing machine in 2" (3 f.u.) and laundry sink (1.5 f.u.) and shower (2 f.u.) and lavs are 1 f.u. apiece. so do the math and increase to 3" once you hit the max of 8.
IMPORTANT NOTE: Tub trap and arm are ok in 1-1/2" but after the vent (within 3.5') it should be 2" pipe for the rest. also will be less prone to clogging.
Q.2. looks pretty ok and should work fine. BTW, how did you draw that? IMHO it came out very easy to understand.