Need to modify bathroom piping before inspection

[SYakoban]

We have a long remodel project including adding a full basement bath. We hired a pro to do the plumbing rough in including under slab and tying into existing piping. At the time, we didn't opt to add a slop sink outside the bath on a framed wall in the unfinished basement side, but want to now. The plumber is not available so I want to modify it myself.

Two areas, I want to be 100% positive about before cutting in anything:

1. The vent.
The first photo shows how I would add in a sanitary T to the 2" vertical coming off of the sink. It's a shared vent with the shower that has a horizontal run of about 15 feet before tying into the main waste stack of the two main floor bathrooms above.

Is that an allowable number of vented items on that 2" pipe?
Do I run 2" or transition to 1-1/2" to vent the sink? If so, at what point do I transition?

2. The drain.
It would have been perfect to add the drain when the rough-in was done due to the under slab location of the pipe, but too late. I want to cut into the vertical drain pipe (shown here) which is around the corner of the slop sink location.

Is this configuration to code?

Any issues to be concerned about?

Thanks guys.

 

[Stuff]

Trap goes under sink, not in wall. You want something more like this: https://terrylove.com/forums/index.php?attachments/img_20150221_105652-jpg.28240/

 

[Sponsor]

 

[hj]

You CANNOT connect the vent like that. Doing so creates an illegal "crown vent".

 

[Atomic1]

I second the crown vent fail. That vent mouth will risk becoming clogged with debris

 

[Reach4]

A potential modification. Note your choice of fittings.

 

[hj]

The "running trap" may not pass inspection either.