Does it have to be unfolded all the way before you can swivel it?

No. The ironing board has two sections. When you fold it down it extends about 22 1/2 inches from the wall. You can swivel it either way and then when you unfold the other section it extends another 18 1/2 inches.

Is the 37" length just the board, or does that include the wall bracket as well?

The board itself is 37" - measured from the wall to the tip of the board is just a touch under 39 inches

What is the width of the mounting plate, and how much higher than the bottom of the mounting plate does the ironing board sit when extended in use?

Mounting plate is 17 inches wide. The height when not In use is 19 inches

Could I attach this to a door?

I doubt it. The door would have to be a solid wood door. The ironing board is very heavy has to be mounted with screws. They make ironing boards that go on doors.