A large enough junction box
First, you are going to need a very large junction box to do all this splicing. Let's count conductors.
- Two /2 cables and one /3 cable = 7 conductors to splice.
- Plus the 7 conductors they'll meet in this box.
- Plus 1 conductor count to cover all ground wires
- Plus 1 conductor count to cover all cable clamps
- Plus 2 conductor count to cover the receptacles
I count 18 total. Let's assume 12 AWG wire, so 2.25 cubic inches per conductor, that's 40.5 cubic inches.
A 4-11/16" steel deep junction box will suffice. (steel makes it immune from nails coming from the other side. This is a deep box.)
Now, to make this thing aesthetic (nobody wants a giant blank plate there), I recommend topping off with a 1-gang "mud ring". That way it will appear to be a normal receptacle.
You're on the right track with "put a receptacle there rather than have a blank plate". However, I'd do a couple things different: First, I'd reroute the cables to a more desirable/expected place for a receptacle, rather than having a receptacle in an odd place for no reason. Second, I'd do it on both sides. Mind you, it wouldn't hurt to make it a little odd; someday these circuits may need maintenance, and you want to hint to the maintainer where these splices might be.
Watch which circuits.
A circuit is "grandfathered" when it only has to comply with the Code written when it was installed, and doesn't have to be updated with each new Code revision. However, there's a fundamental rule: When you're grandfathered, you can't make things worse.
Read the current Code on laundry room outlets. I believe it says the circuit must be dedicated and you can't add outlets to it. That's because when you have separates, the washer is powered off that and the dryer is powered by the 120/240V split-phase. (they make "physically separates" where the washer plugs into the dryer, but they use some sort of internal design features to avoid overloading, since a 120/240V circuit normally doesn't have the capacity to feed both).
If someone hack-a-dacked your dryer outlet to include a 120V outlet, that should be remedied ASAP because it's a code violation. Also the washer outlet is supposed to be 12 AWG/20A (yellow or larger). Also make sure the dryer outlet and cord is NEMA 14 (4-pin); it's obviously a modern installation, but sometimes people fit the obsolete, dangerous and illegal NEMA 10 3-prong dryer outlets on newer houses to match dryers from older homes (you're supposed to change the cord, but builders don't feel that's their bailiwick).
If the other circuit goes to a bedroom, then that is the one to tap for receptacles.
Grounds can go together. Neutrals must obsessively separate
It's OK to clump all the grounds together,(don't forget ground to the junction box if it's metal), but not required.
Separating neutrals is absolutely required. Don't bundle all the neutrals onto one big wire nut!
Actually, (this is a bit OCD) but I might actually use different color wire nuts for the different circuits - red for the #10, yellow for the #12, and orange for the #14 (if it fits). It's sheer coincidence the colors are that way.
Give yourself LOTS of wire length inside the box - a full 12" (including in this case, an inch or two of sheath sticking out beyond the clamp, for cable identification purposes). The 12" will seem ungainly to work with; but it'll go easier if you splice the grounds and push and tamp them into the back of the box first, then (after testing) push back the dryer conductors, then the laundry room (since you won't be tapping it), and finally the other circuit so it's on top.
The mandatory wire length is about 7" - six inches beyond the cable clamp/sheath AND 3" beyond the finished surface of the wall. However you'll hate working on it if you do that, and it'll be a code vio to cut it at all, which is leaving yourself no freedom.
I would not use push-in connectors unless you enjoy really nasty, difficult troubleshooting problems involving guessing where mystery splices are (remember, we are creating two mystery splices here). Push-in connectors, like their "backstab on receptacle" brethren, involve a single, tiny point of contact for all the current to flow through. The manufacturers have gone in backflips to assure this fails safe, i.e. with a burnout/open instead of a flaming arc fault, but the fact it does fail. The other problem with stab connections is they say they're multi-gauge, but who's kidding who? The receptacles said the same thing and that turned out to be a lie, and UL delisted a bunch of 12/14 backstab receps.
Wago style "lever" connectors are certainly a better choice than push/stabs. I'm competent at wire nuts, so I recommend those. The trick with wire nuts is tighten them HARD, and do a vigorous "pull test" holding the nut and yanking each wire. If any wire pulls out, that's an electrically bad connection that will fail, and requires you improve your technique.
Code doesn't care about rubbish splices that fail safe, because the National Electrical Code is written by the National Fire Prevention Association and devices approved by (insurance) Underwriter's Laboratories... not the National Make Your Electrical Work Association and Reliability Laboratories.