Why not just close the door?
For the doors there are two options you can do is either increase the gap to 1.5 inches or use the wall register and bore out the door and install the wall regestier. there are few differnt size you can get and choose from but becarefull when you cut out the opening on the door make sure you don't hit the edge part which it will useally have soild wood to support the door.Baby gate - would rather have option of stepping over for my 2 daughters and I, So do not have to open all the time.
Doors are cut above the floor about 3/4" gap, but with no cold air returns in bedrooms it works better to leave doors open.