The plumber most likely didn't bring it up because (1) he doesn't know much about the height requirements nor is he required to or (2) he is just there to do the plumbing or a combination of both 1 and 2.
Next, basically the general rule is that things like this can be 'grandfathered' in, so long as they aren't touched. Since you were code compliant at the time of installation or construction (or somehow got by), they can not require you to upgrade anything to meet modern codes. However, if you ever go to do any new work or renovation like this, anything that is done must meet the current codes.
I did find, and this copy of the code is old (Seventh edition - 2009), an exception to the ceiling height.
780 CMR: State Board of Building Regulations and Standards
5301.1 - Exception 2:
Ceilings in basements without habitable spaces may project to within six feet, eight inches of the finished floor; and beams, girders, ducts or other obstructions may project to within six feet, four inches of the finished floor.
This exception may no longer be around in the current code, and even with it you might not fit within it's parameters.
(6'-8" overall from floor to ceiling, 6'-4" under the duct from floor to ceiling, but still 6'-8" regardless above the bathroom fixtures)
If you still do not meet this criteria (again, if and only if it's still in the code) then they do not have to permit the work and the job can not be done. I'd suggest talking to the town to see what can be done, they'll give the definitive answer.
Years ago, my wife and I moved into a condo that had a faint weird smell. I put charcoal into a net bag like the ones potatoes come in, and set it into the cold air return ducting at the furnace. I then set the fan to run continuously. We also put the same netted bags in some of the other rooms. After a few weeks the smells were gone. The charcoal absorbed the odors. We built our present house 20 years ago. My wife has a very sensitive nose and detects every odor. I had the furnace guy install a special filter rack where I insert a charcoal filter into this rack ahead of the normal air filter. This charcoal filter takes out almost every conceivable odor that is in the air. Of course, I have run my furnace fan 24/7 for 20 years and counting. I buy mine at Home Depot or Lowes. They have a plastic frame that can be cut to size to fit the filter rack. No odors in my house (happy wife- happy life). This worked for me.
Best Answer
All of this information will be available on most county (.gov) websites through title information or deeds. They only care about when the house was titled, which as @bib has alluded to comes after a occupancy permit and a finished inspection.
It does not matter when the house first started to get built. The inspections will be done when it is done and will require everything to meet code when it is being inspected, not when the work was done. So a house started in 1980 and finished in 1990 and inspected and titled in 1990 is "built" in 1990 for all intents and purposes (given that you do not own a historical home where this information might be important).