It really depends on the situation and every home and need are different. In the basement overcooling situation, you gotta do it for the comfort. Some people have insulation between the basement and living space so you don't want to lose your heating to that area. Then there is the problem with imbalanced pressures. If there are supply duct openings, where is the air going to go? If the basement is closely coupled to the area with a return (upstairs?) and you don't mind a little air down there to dehumidify, or warm it up a little bit, leave it alone. If the basement is sealed off from the return upstairs, it may leave the area more closely coupled to the outdoors. It will be pressurized and the air will tend to leak out through the openings and cracks to nature. That will, in turn, depressurize the living space and suck unconditioned air from random cracks and openings up there. It might even come down some flues. If that is going on, you gotta fix it. You can do it by adding a return or sealing the supplies. Ducting systems are complicated. I hate them.

Do you have flexible or rigid ductwork? How long has this been going on—is this a recent decrease in comfort/airflow or has it always been like this? Do you have the same problem when the system is in heating mode? Take some pictures of anything you can see (without jeopardizing your safety/comfort, of course!), please. It could be something simple, like a kink or an incomplete connection.

My son's new house had a low slope, well shaded roof, with no reasonable attic access for adding insulation or ventilation. The slope was really marginal for shingles, so we went with a rubber roof. We tore off the failing shingle roof, left the existing decking, put the 4" recycled foam panels down in an overlapping pattern (8" total thickness, ca R-42 to 48) and screwed them down with long screws with ~3" washer heads on them. We then applied an EPDM rubber roof directly on top of the foam, exactly as is done normally in commercial roofing. We had to build up the trim at the edges of the roof to accommodate the extra thickness. In a full sun exposed roof steep enough for shingles, especially in an AC intensive climate, I would have put down 1x4 sleepers and another layer of decking on top of the foam with soffit and ridge vents to keep the roof cooler, prolonging the life of the shingles, and decreasing the added heat load into the house. (Even more likely would have been to cover it with solar panels, but his site is too shady for that) Apparently it is standard practice on flat roof commercial buildings with rubber roofs to replace the foam whenever the building is re-roofed. The condition of the recycled foam ranges from badly degraded to like-new. Obviously you want the good stuff. Enough foam to cover his 3 br house with 8" cost about $750 as I recall. I was able to pick it up locally, delivery probably would have been that much again. Installing it was a thousand or two more than the rubber roof would have been by itself. That work, combined with blown-in cellulose in the walls, rigorous air sealing, spray foam around the sill, and rigid foam on the basement walls, along with replacing the metal-framed, single pane windows, dropped his heating/cooling cost from $3-400/month down to below $100.

I am not an HVAC professional. However, I have found doorways to inhibit air movement even when open. IMO, the vents in the workroom and the room with the walk-in closet won't add much heat to the open area adjacent to the library. And their distance from the family room further decreases their ability to affect that space. I would say add the extra duct, but it's going in near a fireplace? Shouldn't the fireplace be providing enough heat for the room? A floor fan can mix the air to provide a more even temperature. IF you find the existing ducts don't provide enough heat, you can increase the return air flow by installing a vent in the bottom of one of the mechanical room doors. Some HVAC people say the space under a door is insufficient for any room larger than a bath.