It is permitted to have a utility tub use the same vent as a washing machine if the tub is next to it on the same floor.

"Critical distances " = a little knowledge, that leads to serious misconceptions. Assuming the red pen shows proposed add-ons, both add-ons are not allowable. All due to the fact that your proposed tub is on a lower floor. The fact that you don't know this already tells me you will need professional help from a Master plumber. Know your limits.

After finding a route for a new vent pipe to go down one floor to the garage, you will still need a Master plumber. My assessment.

Someone may suggest using an AAV. (Search on this term to learn more before posting questions). An AAV works well when the house is all on one level, so there is no positive pressure. AAV's work to let in air under negative pressure but do not compensate for positive pressure. Other disadvantages too. One is that being mechanical they may fail one day.

Take nothing written here as The Word. Knowledge presented in Internet posts does not replace real in-situ inspection and advice from a trained professional.

hth

I just commented on this on another posting of this on this forum. Anyway, rerunning the venting 12' over and teeing into another vent is legal, however teeing in the laundry tub in the garage into that stack is not, if it is not vented. A separate vent pipe would have to be dropped to it. And the AAV's are not legal, at least here where I am, for the reasons davidro mentions. And like he says, If all else fails, call a plumbing professinal like myself.--The Captain

The critical distance I was referring to was the distance from the trap to vent for each fixture was okay, and would not come into play as a problem. I can find very little about special requirements for washing machines, which was why I asked.
So supposing, I can somehow find another place to run an auxiliary vent up from the sink and tie into the vent for the washing machine, if the washing machine vent is now an auxiliary vent too can I have one run into the other, or must I keep the washing machine vent as is, since it now has an auxilary tied into it.

Hope that was clear.
My other option is a Flotec Utility sink pump that requires no vent or trap. That may be my easiest option anyway.
Thanks

kirsh: there is no 'special requirement' for a wash machine, it should just require a 2" stand pipe and trap and the pipe running from the trap should be 2". The trap-vent distance, at least here in Minnesota, is 60". so in other words, from the 2" trap to where the vent tee is can be no longer than 60". The vent only has to be 1 1/2", (the new vent in red in the drawing). For your new laundry tub in the garage (red line coming from the laundry tub in the drawing) that line only needs to be 1 1/2". It needs to be vented as well, with a tee for the vent somewhere between the trap for the LT and where it tees into the pipe coming from the wash machine up above. The max distance from the trap on the LT to the tee for the vent is 42" on 1 1/2". The vent pipe for the LT only has to be 1 1/2". You would have to run this vent up the wall next to the 2" pipe coming down from the wash machine, and tee it in about 42" above the floor. At least, that is how the code reads here in MN. I wouldn't mess around with a pump when you have nearby plumbing that would work. Hope this helps.--CaptainBob

Thanks for the tips Captain Bob. I tore the wall off on the garage side today. I am in luck. I can plumb the 1.5" drain for Laundry Tub and actually tie it in under the house to the 2" drain coming from the Washing Machine so I can have the stub low enough for the LT p-trap.

The LT vent I can run up the wall , but I will have to run at a 45" angle once I get up a couple feet from the tub. It will go straight up again before it penetrates to 2nd floor. This is OK , correct?

Once the LT vent is in second floor, I can tie it into WM vent, provided this is done 6" ubove flood rim (i.e. top off WM standpipe.) thus the LT vent is considered an auxiliary vent.

So, I can then run the WM / LT vent up the second floor wall into the attic and tie into that other vent about 12 feet away. This is where I am unsure. Can I have an auxilary into auxiliary?

Then I can abandon the old 1.5" WM vent.

I am also wedging in a WM oveflow pan drain too. That will be fun. That will dump into laundry tub, w/ 1" AG. That will have no p-trap or vent as I understand, since it cannot be tied into main drain system. Maybe a check valve where it exits into sink, to keep the bugs from crawling up? Not sure how big that line needs to be.

The picture should make all clear.

Kirsh: Should work, according to your drawing. It is okay to have those 45's you are talking about. Tee it in to the wash machine vent like you said, at least 6" above the top of the wash machine stand pipe. I always recommend the tee at 42" off the floor, usually the top of a washer stand pipe is 36" off the floor thus 36" + 6" = 42", but make sure whatever everything is at you have at least the 6".
And, yes, you han have that "auxillary" vent tee into a vent that is teeing into another vent. Many houses have just one main vent going through the roof and all the vents for the fixtures all tee off of one another.
The old WM vent you are abondoning cut it off and cap it close to the tee if you can. I think it looks bad to have old pipes sticking up that do nothing.
The overflow pan isn't a bad idea. Like you said, it doesn't tie into the plumbing system, just have it indirectly drain with no trap or vent into the laundry tub with a gap above the rim of the LT, at least an inch or more. The line can be whatever size you want, I've seen some that are 1", the ones I use have an 1 1/2" opening. 1" will be fine. It may never get used, but would be a problem saver if it does. I wouldn't recommend a check valve on the end of it, just something that could get stuck and then defeat the purpose of it. I've never seen a critter problem with these indirect drains that I can remember. Have fun and good luck--TheCapt.