Maineman/Ont girl

The medical determines the validity of the visa.
After you've had your medical, you have 6 months to complete your interview AND enter the US. How the two latter events are spaced are up to you. For practical reasons (giving time for visa to be issued and delivered), I would not recommend having the medical more than 4 months prior to the interview.
Example: I had my medical on March 15th, my interview April 28th and had to enter the US before my visa expired on September 15th.
If you have your medical on December 22nd, your visa will be valid until June 22nd (or shorter), regardless of whether your interview is in January or March.

Hopefully someone who knows for sure will chime in, but I'm going to say I don't think it would work that way. When I was getting my certificate at my local station, there was a woman from out of town who wanted to get her certificate there that day. They told her they couldn't do it for her and she would have to go to the police station in the city where she lives. So, based on that, I'm guessing no.
(weird though... since they are searching the database for all of Canada, it shouldn't matter where you live or which station you goes to...)