If it were me, I'd install a new sanitary tee in the main drain, just above the existing sanitary tee. I'd have the standpipe for the washer come down into a P-trap, and connect the other end to the new tee.
However, you'll have to make sure the standpipe is at least 18" above the trap weir. If that height cannot be maintained, you'll have to move some things around.
I'm also not sure if a proper P-trap will fit in the stud bay, so you'll have to measure to verify.
The minimum length of a trap arm is two pipe diameters. So if you're using 2" pipe, that's 4" from trap weir to the sanitary tee.
Then I'd have the trap from the washtub, connect to the existing sanitary tee.
Basically, something like this...
![Drains](https://i.stack.imgur.com/0HUVn.jpg)
![Minimum drain lengths](https://i.stack.imgur.com/asD9w.jpg)
You can't tap a cable like that - NEC has a whole section called the "tap rules" which describe when you can do that, and this is not one of those times. Even if you could, it would be a lost cause: Since a washer must be provisioned for 20A, that plus the 30A dryer puts you at 50A, and you'd need to run the expensive 6/3 cable.
Here's what you can do:
Pull a 12/2 cable alongside the 10/3 cable
That may seem stupidly obvious, but for the cost of a 12/2 cable, you are done. It will take an additional breaker space.
Run 6/3 and install a subpanel
You can run 6/3 cable instead of 10/3, terminate at a small subpanel, then have a 2-pole 30A breaker for the dryer and a 1-pole 20A breaker for the washer. I believe the subpanel would need to be at least 50A.
Run 10/3 to a subpanel with a "generator" interlock
Use the interlock so you can only operate the 30A breaker for the dryer, OR, the 20A breaker for the washer. You would have to throw the breaker every time. Very tedious. This will get old fast.
Buy a washer-dryer set that's designed for that
Large housing complexes are typically out to minimize the utility work they must put in laundry rooms. Industry caters to them with washer-dryers designed to run together off a 30A/240V receptacle only. They are either sized correctly, or have logic to manage electricity usage so they can't overload the 30A circuit. They make combo units (washer + dryer in one unit) which tend to be tenant-tier cheap in their feature set... or they make matched set washers + dryers which can be deluxe, but expensive.
Best Answer
First, congrats on the very big leap up, skipping clear over the "housemate shares and buying used stuff on Craigslist" phase of young life.
That space is not intended for stackable separate washer/dryers, which are a recent "fad" that are a side effect of the trend to front-load washers (also a deluxe item).
That space is for pre-built washer-dryer combos, which are simply a washer and dryer pre-stacked onto one chassis, like this unit. These units are a stock-and-trade of rental units and condos. Most are small capacity to make them narrow; large capacity units can be had.
Note the wedge opening that allows access to the top loading washer on the bottom...
Lately, however, the new trend is for "washer-dryer all-in-one" units, which seem to defy the laws of physics! They place washer and dryer function in a unit the form-factor of a washer. Put clothes in dry, they come out washed and dry. All the rage on tiny houses, boats, etc. I'm skittish of this one because of its low price, but there you go...
Dirty clothes go in, dry clothes come out...
Keep in mind ordinary wear and tear should not be counted against you, as a result many tenants get their entire deposit back. But you do not have any legal right to modify your apartment, and you will pay the full cost of restoring your apartment to status quo ante, which will be billed to you if the deposit doesn't cover it.