Your pictures of the existing plumbing, like hj has mentioned are a bit obscure.
We don't have a clear picture of what you intend to do.
For these reasons, the pros tend to skip over it.
Are you really a licensed electrician? I would expect more respect for building codes if you were.
The cost to do it right, and the way you have suggested is going to McDonalds for lunch a few times.
We're not talking killer amounts of money. It's more about "choosing" the correct fittings to assemble that mess. Or, it wouldn't be a mess if hj or I had done the work.
Distance from the trap to vent
The lav can wet vent the tub, as long as that is within 42"
You can trap arm over the lav a combined 42" distance too, so if it were installed locally to both, you could trap arm over and catch both.
Some codes don't care about a toilet vent, but I like them for several reasons. For one, I like the water in the bowl to be calm except when tempted by flipping the flush handle.
If you choose to vent the toilet, a wye fitting would be nice.
Someone mentioned Bert Polks pdf on plumbing. It's killer information, and should be studied by anyone that wants to think like a tradesman. Kudo's to Bert!