This whole project is going to be way beyond affordable. It should be a teardown, as that will be cheaper than all the alterations you want done. A teardown will also be a much better end result than leaving the questionable, old, cramped, first floor underneath this not great design.

Go to a local flooring company, any carpet can be cut and bound to a rug in a custom size.

I would definitely prefer the first layout. First, the room with windows on three sides would be my favorite room in the house. Second, I think with your busy family that your island should move toward the fireplace a bit to give more than four feet aisle space. Third, that ten foot wide space will be a busy traffic area from the hall into kitchen and living room and won't feel too big in my opinion.