That existing 4' high loop is really delving into a rather gray area of the code. Normally the trap is considered the official point of transition between an appliance drain line and the house DWV system and in that case the manufacturer of the appliance may determine the required diameter of the appliance drain line which is determined by the volume and velocity of the discharge from the appliance. Generally the manufacturer will print a table in the installation guide that defines the size, length and permissible vertical lift of the drain line with recommendations to increase the line size if you need to exceed the length or vertical lift. That table will also define a maximum length or lift which is determined by the output pressure of the pump in relation to pipe friction loss or maximum vertical static head. In most cases the code will require that the appliance drain must be in an accessible location to permit servicing in the event that line should become obstructed. Whenever a drain line is permanently installed in the wall it is then classified as a portion of the structure DWV system and it would require that there must be a trap at the beginning of the concealed section of the line and the minimum permissible diamter of the line would then be determined by the DFU (drainage fixture unit) load value of the appliance or fixture being served by that drain. In the case of a dishwasher the DFU load is 2DFU's which would require a minimum of an 1-1/2" drain line. The plumbing codes also require that a dishwasher, washing machine, HVAC unit or a home water treatment discharge line must also be connected by means of an "indirect waste receptor". Under some local codes you could achieve the indirect waste receptor by installing an "Air Gap" or they may require you to install a standpipe on the trap and connect the dishwasher drain to a standpipe in the same manner as a washing machine line is commonly connected to a standpipe. (Some codes require all dishwashers to have an air gap, regardless of how the drain is connected. Check your local code.) Under the UPC whenever we have two common vented gravity flow drains connecting to the same vertical riser we are required to use a "Back to Back Tee" (4-way) however code prohibits connecting two commonly vented drains by means of a Back-to-back if one of those drains is receiving the discharge from a powered drain I.E. a dishwasher or washing machine. The discharge from a powered drain is required to be connected by means of a directional fitting (baffled Tee) and generally the baffled tee should be placed lower than the gravity drain. In your case the simple solution would be to connect the dishwasher to the disposal input port, providing that is permitted by your local code as well. (For reasons known only to the AHJ some codes prohibit connecting a dishwasher to a disposal input port..One code that immediately comes to mind here is the Illinois State Code.) Your screen name is Tom-in-Seattle so I am assuming that you are actually in Seattle. I am not certain but i believe your code is modelled after the UPC (Uniform Plumbing Code). If so the defined trap and waste arm size for a kitchen sink is 1-1/2" but under the UPC all vertical risers and drain lines serving a kitchen drain must be a minimum of 2" pipe. You may use an 1-1/2" pipe for the vent. There is some question about whether the length of the horizontal offset on the vent in your attic would pass, but I doubt that it would even come into play unless you happen upon a hardnose inspector that is really intent on finding a problem.

There is no mandate for a 4" vent line. The codes have tables that assign a Drainage Fixture Unit (DFU) value to every type of fixture that would be connected to the drainage system as well as define the minimum size of drain permitted. We begin by computing the total DFU load on the structure then refer to another table that defines the maximum number of DFU's that may be handled by each size of pipe. Because of the size of the anticipated solids in the waste a watercloset is required to have a minimum of a 3" waste line thus the branch from the watercloset and the "house main drain" must be a minimum of 3" but for large structures with multiple bathrooms it may require a larger line for the main drain. Under the International Residential Code all structures are required to have one "Main Vent" which must run undiminished in size from the "House Main Drain" through the roof. When additional auxillary vents are required they may be reduced to 1/2 the diameter of the line they serve and they may terminate through the roof or in the attic space. In some instances subject to a number of restrictions they may terminate directly through and outside wall or you may use an unlimited number or air admittance valves. The Uniform Plumbing Code does not require a Main vent. Under the UPC a vent line may be reduced to 1/2 the diameter of the line it serves but the combined cross sectional area of all vents must be equal to or greater than the cross sectional area of the main drain line. Under the UPC ALL VENTS must terminate through the roof except, the UPC will permit a maximum of one AAV per structure with the expressed approval of the local authority having jurisdiction. In plumbing any line that is vertical or angled not more than 45 degree from vertical is considered a "Vertical Line". Any line that is horizontal or angled at any angle up to 45 degrees from horizontal is defined as a "Horizontal line or or Flat Line". A vent line may not be run horizontal until it reaches an elevation at least 6" above the flood level rim of the highest fixture served by that vent which would be the lavatory sink rim. It might be possible to make the offset inside the wet wall in the room below by making a 45 deg angled offset however the length of the drain lines from the tub, shower or lavatory trap to the vent is very closely regulated. Offsetting the vent below might extend the distance from the trap to vent beyond acceptable limits which would then require an additional auxillary vent for that fixture.

Goods news about having the water source really close. That's half the battle. Except for having hot water & that would be relatively easy to fix with either a small electric H20 heater or a demand use one. Not so sure about the gray water from a kitchen. You may get into problems with codes, etc. Having a septic tank could possibly make the sewer problem easier. How far is the septic tank from the proposed new kitchen? With the sink on an exterior wall, maybe you could just run a whole new line to the septic? This is sounding more & more do-able & shouldn't be hugely expensive, either. Esp. if you're dealing w/ a modest distance to the septic...or to a point where a new line from the kitchen could tie in w/ existing line. Are you in south Texas? If so, that also makes it a lot easier & less expensive because of either having a very shallow or non-existent frost line. (You could also explore locating a small batch water heater in a closet on the exterior of the house if interior space is at premium?) Good Luck! I'm anxious to hear how this turns out.

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.