Yes.

I would leave the door as is and replace the flooring with something that coordinates with the door. If you must do something - stain, don't paint.

Yes, I would.

The door that is hanging will be replaced by the fir one with glass.

Love the new door, would look great in my house.

With little info I would stain it if the wood is nice but with really no context hard to advise. You had a stained door before so are you thinking the same stain color ? If so and you loved it that is the right choice

It's a bit difficult to see in the photo but staining the door to be closer to the flooring would probably look better than leaving it as is.