Yes, in some areas, the space is mandated by local codes.

And the HVAC system/set up does make a difference.

The codes have to take into consideration worst case scenarios---in other words, doors always closed.

What I have done for some people is to remove the door, cut a dado(groove) in the bottom of the door, and install a strip of wood in the dado that fills most of the space. When that strip is finished the same color(or close) as the door, it virtually disappears.

However, when I did that---I warned the people the HVAC operation would suffer and they may want the strips removed. Some did.

The codes have to take into consideration worst case scenarios  in other words, doors always closed.

And they really need more than half an inch to avoid that? I would understand it, if some clearance were needed on top of the door  for the hotter air to escape towards the central A/C return. But what good is the clearance on the bottom, where only the cold air (that's pushed into every room through each room's own opening) can escape? What I have done for some people is to remove the door, cut a dado(groove) in the bottom of the door, and install a strip of wood in the dado that fills most of the space.

Most interesting  thanks!.. How much should this cost per door?

This is a matter for the HVAC designer rather than the building code or the GC.

You might want to try the air conditioning and heating forum.

I used to charge $25 per door, 3 or more. 2 or less was $40 per door.

The space is at the bottom since the cold ait returns are located at the base of the wall---by code.

"The contractor has hung them all with rather large clearances between door's edge and floor  partially defeating the sound-insulation... When confronted, he pointed out, that Code mandates clearance between 5/8" and 7/8" in our case, where the central air system pushes cold air into every room, but has only one return per floor (in the middle). "

There is absolutely no such mechanical code "requirement" that there needs to be a minimum door height above the floor for AC or forced air heating purposes.

NONE.

What you have been told is complete and utter NONSENSE.

I am a nationally certified Residential and Commercial Mechanical Code Inspector and challenge anyone to provide a Code section from any Code that has ANY such a requirement.

While a design to provide a balanced air output and return air within a forced air system IS required, the 'means' by which this should occur is written into NO CODE and depends upon the design and the circumstances.

Particularly on older existing homes with centralized (not individual room) air returns like the original poster's...the house should simply not ever be 'tight' enough to restrict air changes per hour or to restrict air flow when interior doors are closed.

Those who think doors need to be cut extra high above floors to maintain air flow are simply ill informed.

There is nothing in any Mechanical Code that prevents even sealed door units and sealed thresholds from being installed.

Nothing.

In some situations, Mechanical Codes actually require sealed doors....(like between bedrooms and closets containing boilers, furnances and water heaters).

Likewise...some rooms, like bathrooms and kitchens, garages and some types of basements and crawl spaces are prevented from having return air vents and ducts installed within them.
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I don't believe your contractor is lying to you. He does not stand to benefit enough for his false claims.

I DO beleive he just doesn't know what he is talking about, and thought he 'learned' something along the way that simply is not true.

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Tell him to repair the doors he damaged at his own expense....and do not pay him until he does.

Instead of coming here and asking everyone their "educated? opinions", why not go straight to the inspector and ask him?
If I have learned anything in the past thirty or so years it is that there "is no absolutely" that will cover all situations everywhere. Cities, counties, states, etc. change things in their code books to "improve" codes that they think are not adequate for their area.

Residential mechanical codes do not usually address the number of return air duct grilles in rooms but they often do limit the amount of air pressure differential between the individual rooms and the central spaces of the house.

No doubt making the door undercuts larger is an attempt to meet such requirements but it is not the best way in terms of privacy. Assuming it is too late to add return air ducts, add some insulated by-pass ducts over the ceiling or through closets (with lots of bends) so the doors can have their undercuts reduced in height or saddles added.

The building inspector should not be relied upon for quality control; your HVAC designer should provide that service. If you let your builder design the HVAC system you made a classic homeowner mistake that you will probably have to live with. Good luck.

Here is a link that might be useful: return air ducts

I have linked to a manufacturer's technical note that explains the issue of door undercuts for air return in residences.

Here is a link that might be useful: door undercuts for return air