Laundry room plumbing question

I want to make better use of the space in my narrow laundry room. The washer water supply and drain come up through the floor. The washer sits against an outside (garage) wall. I want to put the plumbing into the stud cavity, to allow the washer to sit tight against the wall, gaining some space.

Can I put plumbing (supply and DWV) in an outside wall? Something tells me this is not OK, but are there solutions to allow this to work?

An alternative solution is to move the washer to the other side of the laundry room, where the plumbing could be installed in an interior wall. But since I want to use a stackable washer/dryer, this will mean the dryer vent needs to move, requiring a new penetration in the exterior masonry wall, which I don't want to do.

Any alternatives on what could be done would be appreciated. Thanks.

Does it freeze where you are? Tucking it into the wall means closer to the cold...you feeling lucky? Wait for the pros who know the codes, but I don't think this is a problem, maybe just not a great idea (again, depends on where you live and the specific wall).

washer

ALL washers have hoses, etc, behind them that keep them from going all the way back to the wall so if anything you may gain an inch or so, unless you cut the wall away so the hoses can also go into the wall.