I've designed and constructed many underpinning jobs in Philadelphia, For Philly buildings with rubble stone foundations, 4' is pretty typical for pier widths. Going 1' below the bottom of the new footing is not unreasonable. You should not need smaller piers at the corners (I assume you mean each end of the existing wall).
Since the existing walls are multi-story party walls, they are tall and heavy. Therefore, you will probably need continuous underpinning piers for the length of each party wall. Another reason for continuous piers is that you are not going very far below the existing rubble stone wall with your new footings. Therefore, your piers will probably be bearing in the same type of soil as the existing rubble stone walls.
I expect that your new building will have its own footings and columns and that the new floor slabs and roof will be supported on their own framing system, independent of the existing structures. If so, you will actually be reducing the loads on the existing walls - except possibly for the addition of drifted snow as JAE mentioned above.
Usually, adjacent underpinning piers do not have or need connecting rebar dowels or formed shear keyways. However, in Philadelphia, one particular structural engineering firm always insists that dowels and or shear keys be installed between piers. This engineer also insists that pressure treated lumber be used for all pit shoring and lagging boards. Reinforcing steel is almost never used in underpinning. The stresses in the underpinning concrete are usually very low and rebars make it hard to dig the pits and allow a worker to enter a deeper pit. Underpinning pits are also usually hand excavated, especially when underpinning rubble stone walls.
Philadelphia has at least three experienced underpinning contractors working there - Moretrench Geotec, D'Angelo Bros., and Schnabel Foundation Company.
Make sure that you do a pre-construction survey of the existing adjacent structures before starting any excavation or other work. Also, monitor on a regular basis these adjacent structures so that you know how much movement they may be experiencing. This will also be very important if the buildings see any damage during the work.
Finally, don't attempt the underpinning unless you have an experienced underpinning crew. Many contractors say they do underpinning, but there are very few who really know what they are doing. The rest just dabble in shallow underpinning and are accidents waiting to happen , especially with deeper underpinning projects. The company doing the underpinning should be doing the pit excavation and shoring, pouring the pier concrete, and doing the drypacking - in other words, the whole works. Underpinning is not work to be parcelled out to several different contractors. When this happens, no one has full responsibility - or authority.