How would you install a 9" piece of baseboard on a badly curved wall?

If the top width of the baseboard is wide enough, I would scribe a pencil line along the top edge that matches the wall bow, transfer the scribed mark to the bottom edge as well and shave off the back side of the baseboard with a power planer or drum sander so that the baseboard back side matches the curve of the wall....or at least makes the gap narrower to fill in with caulk. It may take some time but due to the short length and height, you cannot get the board to flex. Notching the backside works in some cases too to make a molding more flexible...cut a series of parallel notches or grooves in the back side about a 1/4" apart with a table saw to make the board more flexible/bendable. Don't cut all the way through to the top edge...leave the cut just below it so that the top edge does not show the cuts. Make the cuts as deep as possible without going through the face of the board or any grooves/molded contours in the face of the board. Or cut the board in the middle of the backside so that it can bow more....set the table saw depth a fraction of an inch below the thickness of the base molding.....cut in the middle of the backside which allows the board to bend slightly.

Confused about the vent issue....if the base does not go under it, I would mark and cut out a section of the base so it could fit. Cut the base to length between the door and wall....then measure, mark and cut out a section where it has to go around the vent. Or just run the base up to both sides of the vent if the bottom section is not too noticeable or can be filled in with a small piece of scrap.

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RE:

How would you install a 9" piece of baseboard on a badly curved wall?

The gap between the center of the baseboard & the wall is about 1/4 inch. Too much to hide with caulk. Because the base is so short, but tall, bending it into place is also not an option.

Also, how would you install it around a vent that is too close to the floor for the base to go under & too...

Curved Baseboard

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I would employ a skilled professional finish carpenter. Not a framer rather a finish craftsman or cabinet maker.

just get some drywall mud and flatten out the wall, strike it off with a straight edge, prime and paint then install the base