You are looking at installing a stud wall along as well as under the stringers. I suppose you could go without walls on one or both sides, depending upon your plans, but I would put a stud wall on 16" spacing under each stringer to box in your stairs with drywall. Think about access panels/doors/drawers for under stair storage. Solidly nailing/screwing in wood risers at the back of each tread should stiffen them up just fine and take care of your bounce. An extra stringer down the middle will fix it for sure. Tear out of the stairs would only depend upon whether you want a finished wood stair and you don't like the wood or tear out wouldn't be necessary if you are going to use carpet. You want 3/4" solid wood treads and not plywood, etc. And the max rise you want is 7.5" and 6" to 7" is better and the minimum tread depth is 11" and 12" to 13" is better. Short rise and deep steps is better for the elderly, etc. to walk up and down.