The vertical wall seams crack more readily because the house shifts and moves. Ive tried grouting only and going back to caulk with sanded grout caulk or silicon only after a crack develops, hoping in some cases it will not. Sometimes it does not crack. Some contractors caulk all seams as a matter of course, no callbacks, safer. I liked the idea of sanded grout caulk to match your grout color It has not held up for me in wet conditions, such as the top of the bathtub where the wall tile starts, etc. , definitely not along the shower floor where the wall tile starts. It has held up in other areas though, and looks nicer when it all ties in. I agree, mildew on silicon is unsightly. Every bathroom has different conditions, including ventilation, moisture, etc. Some people have to clean out and recaulk the silicon every year or two. Whether its allegedly mildewproof silicon or not.

Remove the grout from those areas and replace with caulk. You can find color matched caulk in box stores, or you can order it online. Since there seems to be a bit of flex in your tub, you could fill the tub with water prior to caulking. That can sort of "preload" any movement that might occur in the joint. Not a requirement since caulk is flexible, but it can help in certain circumstances.

Yes, the plumbing needs to be done first. Yes, you may be able to just get another acrylic shower wall to cover over everything, if the wall is not moldy or wet behind the tile that you are looking at. There is no need to replace the tub itself if it is in good shape, even if you do decide to take down the wall tile and replace everything from the studs out. Just make sure the workers cover the tub well to protect it while the work is being done. I am with Monicakm_gw, get some more bids for the work. You might want to have them bid on just replacing the walls with new fiberglass, and on doing the whole new tile job, too. You can see new walls at Lowes, Home Depot, and Menard's online. There is also the whole Marbleite wall option to look into. Local dealers for Marbelite are supposed to be cheaper than buying from big box stores, if I remember right. While you are fixing plumbing, you should probably put in a proper mixing valve to protect from scalds. It is the code now. No one will be able to turn the water too hot to burn themselves and flushing the toilet or using the water elsewhere in the house will not burn the person in the shower anymore, either. I have read the experts here on the forum say that the Symmons Temptrol system is very good and is a good value. Here is one that comes with the shower head and tub spout for under $100. http://www.homedepot.com/p/Symmons-Temptrol-1-Handle-Tub-and-Shower-Faucet-in-Chrome-S-96-2/100643619

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.