A. Possibly, but each municipality will have its own procedure to give you one. Having a placard on the rear view mirror serves little purpose since the restricted traffic zones in the town centers (ZTL or Zona Traffico Limitato) are enforced by camera only, which takes photo of the license plate. Therefore the license plate number must be registered in the individual City White List, on the respective cities' portal (I don't think there is a national database). If a plate number is not registered in the white list as "authorized" the fine is automatically issued by the system. There is no human interaction with a police officer to which you can show a placard.
B. Each municipality will have its procedure, and it is generally available online. Unfortunately I don't think there is a national database, therefore you would need to do so for each after you know the license plate number of the car you rent.
C. Yes you can enter the restricted ZTL and park in designated handicapped stalls in the restricted zone, but only after the license plate number has been authorized in the respective City White List, in the City Portal. That can be done also in person at the Municipal Police of each city. As I said the process is enforced by cameras, and cameras can only see your license plate, not the placard hanging on the rearview mirror.
D. Besides what specified above, the other options is park outside the city centers at any of the parking lots and rely on taxicabs, where available. But I don't know about your level of mobility. Italian historical city centers are ancient, built long before cars were invented. Besides traffic restrictions in the historical cores, there are also streets that are so narrow, or even involve stairs, which would not be accessible even by cars authorized to travel inside. In some cases walking is the only option.