I think nobody has answered because that is just too wide open of a question. Moving drain/waste/vent pipes is generally straightforward, but there are many many things to consider, such as proper slope in the drain pipe, proper number of bends, proper placement of cleanouts and wyes, etc. Your sink definitely should have a vent. It's possible that the vent pipe runs up from where the pipes from the sink enter the wall and then horizontally to a vent through the roof somewhere, but that seems unlikely. Running a pipe outside the wall even for just 5 feet screams "non-professional" to me, so I wonder if perhaps the vent was never installed. You'll have to open the walls to see.

The builder was over and I asked about the big stub. He said it isn't for a toilet. He said it's a clean out valve. He said I could cut it down so it's at the floor level, but I can't see that being smart. If I cover it with a floor, I won't have access to it anymore. That's why I want to move it. I wanted to start on here before I called a plumber to get an idea of what I'm getting into. I also live in the boonies so I can't see a plumber coming out without charging me for the trip. I won't even be calling one if the cost is going to be ridiculously high. I have no idea if I'm looking at $1000 or $10,000.

"Any ideas on what I should do?" Sewer gas isn't something you want in your house. Let me say the obvious... perhaps you should invest in someone giving it the once-over and replacing it if needed. It sound like it may have been a "creative" installation at the time it was done. 50 years isn't that old for cast iron... but it is for a can of tuna. BTW: Another tip from Mr. Obvious... Don't eat the Tuna!

This long and narrow bathroom reminds me of the one I grew up with in Brooklyn. It had the same problems you are facing. I highly doubt there is a radiator in this bathroom. The OP can comment if it exists or not. The large diameter steam pipe in the corner is a typical set up for bathrooms in old buildings. There is probably one thermostat controlling the heat for the entire building. The apartment above does not get to set its own temperature. How warm you apartment is in the winter is determined by what floor you live on. I agree if this is the only bathroom then it should to have a tub. I am not sure if re-configuring the tub is going to lower the resale value on a $1M+ coop in NYC. Putting up the wall between the tub and the sink makes the shower more functional. But it will make the bathroom look smaller and block some of the light from the window. I still think if you are going to move the tub the something like this one at Home Depot could work. It is 6 inches shorter than a normal tub. You will need 54 inches from stud to stud which may be possible once the plaster and lathe are removed from the walls. You could install the tub 8-10 inches away from the window wall. A shelf could be built between the tub and the wall and covered with tile. That becomes a place to hold your shampoo bottles and soap dish. Most of the steam pipe would still be exposed and will keep the bathroom warm even with the shower curtain closed. The shower head and controls stay on the same wall. This will open the space to put in a 36 - 48 in vanity with a granite counter top and a wider medicine cabinets and better lighting. That will give you ample storage space. Something that was not an issue when this bathroom was laid out 100 years ago. Have you asked any of your neighbors if they have remodeled their bathrooms? It would be helpful to get ideas and see how they dealt with these challenges.