Plumbing utility sink into washing machine with no vent
Don't stack the san tee's on top of each other like in your second drawing. As the top fixture drains it can create a vacuum inside the Lower trap  arm and siphon the water from the trap allowing sewer gas into the room. What you want to do instead is use an upside down wye or "Y" fitting (so there is two openings on top and one on the bottom)  with a 45° elbow on the branch that isn't completely vertical.   You should end up with what looks like  a lower case h but flipped upside down. Now that you have two separate vertical lines you can add your san tees on top of each, then run your drain line to your sink,  and the other can serve the laundry.   

And as for the vents.   On top of your san teesRun the pipe up higher than the flood rim of either fixture (the point where water will overflow), so either top of sink or washer whichever Is higher.  Put a 90° elbow on one and point it towards the other pipe.  Take one more san tee but flip it opposite direction (the middle hub/opening should curve upwards not down) put that on your other pipe pointed at your 90° , tie them together.   Add another small piece of pipe to the top and a AAV Or studor vent on that and your all set.   

The condensate and air conditioner you can run to the corner of the sink,  or you could put another upside down wye &amp; 45° on the laundry standpipe/drain ABOVE THE P TRAP and run them into that.  

In that situation they will act as a sort of trap primer.  Not that you would need one there.   But the important thing about those two lines are that they must have a 2" air gap.  Meaning you can't tie them directly into a sanitary sewer line.  They have to be an indirect drain. 

BUT*
This has all been with the assumption that your local authority having jurisdiction uses these Uniform Plumbing Code.  I can't speak for IPC however.

And for any of this to be true  you would need to know if the stand pipe has a p trap below the floor or not.     If it does you shouldn't be adding any fixtures to that stand pipe.