Your descriptions are a bit confusing. In the text with the first picture you stated that the 2" on the upper left side is the vent, while the 2" extending to the right is the drain line from the lavatory (bathroom sink). If that is correct, then both of those lines are illegal.

If that were in fact a vent line with nothing else attached it would be illegal because code prohibits any horizontal run of a vent until the vent reaches an elevation at least 6" higher than the flood level rim of the highest fixture served by that vent.

If that 2" line is in fact the drain line from the lavatory it is also illegal. The code mandated size for a lavatory bowl drain is 1-1/4". The code will permit increasing the size of a line or trap by a maximum of one nominal trade size, which would then be 1-1/2" so 2" is again illegal. (Note: the lavatory bowl is the only fixture in a house that is permitted a 1-1/4" line. Most plumbers now opt to use the code exception and install an 1-1/2" trap and line to preclude the necessity of maintaining a separate inventory of pipe and fittings for this limited use. In fact, installing the 1-1/2" trap and line have become so commonplace that most hardware stores and home supply centers no longer carry 1-1/4" pipe and fittings, although they do carry the 1-1/4" traps and trap adapters)

Under your second picture you state that the 2" to the right is the drain line for you bathtub. If that is the case, then am I correct in assuming that the line on the upper right is connected to you lavatory? If that line is connected to the lavatory and if there is a vent line extending upwards from the lavatory location your vent is correct. The portion up to the lavatory tee connection would be a "Combined waste & Vent" and it may have a horizontal section.

Now in regards to your fittings;

If that riser on the left is going up to the lavatory then extending up to the roof for a vent you may install the 2x3x3 sanitary tee as you have it laid out.

The Wye fitting for the tub drain should be a 3x1-1/2x3 Wye and the side opening MUST BE configured so that the side opening in on the side of the pipe. If you are under the IRC you could then have up to 6Â of 1-1/2" pipe from the Wye to the tub trap without requiring any additional venting, or you may opt to use the 2" that you have in which case the line could be up to 8Â from the Wye to the trap. (If your under the UPC a 1-1/2" line may extend 3Â6" or a 2" line may extend 5Â without any additional venting)

The 1/4bend under your closet flange is correct and the water closet will not require any additional venting.

I reread my description and yes, it is a bit confusing. What I state in the 1st picture is correct, the line to the right is the drain for the bathroom lavatory, not 'a bath'. It originally was for two lavatories, one on the 2nd floor that drains down through a 1-1/2" section and then merges with a 2" section where the lavatory on the 1st floor powder room tied in. This then continued down through the floor into the crawl space as you see with a 2" line. It sounds like this should now be changed to 1-1/2" pipe since it will only be carrying waste from the upstairs lavatory.

The 3" stack you see in the original post goes up through the floor to the closet flange for the 1st floor powder room (just out of site in the following picture). As far as I can tell that 2" pipe on the left in the original post coming off the 3" stack attaches to a 2" vent line, which is the line on the right in the following picture.

The 3" stack in the above picture is the drain for the upstairs toilet. It merges with and drains into the 3" stack below the section shown in the original post. At the top of that stack it also has a 2" takeoff that I believe merges into the vent line. Nowhere is there a continuous 3" stack than runs up through the house.

I'll post a picture tomorrow of the existing closet flange and the location where I want to move it.

Here's another picture. This shows the existing location of the closet flange, which drains straight down per the 1st picture in this thread. The new location is indicated by the new flange.

I'm deleting the lavatory drain and vent line and moving the drain to the wall on the left.

Of the pipes shown in this picture, the 3 inch on the left is the upstairs toilet drain, which merges into the drain piping for the flange shown here. That merge is out of sight in all previous pictures. The next pipe over is a 2 inch vent that attaches just below the existing flange, the next pipe is the vent for the lavatory and the one on the right is the drain coming down from the upstairs lavatory. It's 1-1/2 inches merging into 2 inch, which then merges with the existing toilet drain as shown in previous pictures.

I hope this makes it all as clear as mud :-)