I would definitely use cardboard to stiffen the shells. Something thin that you can sew through like poster board or pieces of a cereal box. Obviously, you won't be able to throw the shells in the washing machine, but I have many bags and wallets that I've interfaced with cardboard and they spot clean just fine.
If your leather punch won't work, it's probably just dull. I've purchased leather punches from a bunch of different places and all were dull when I got them except for the ones I got from Tandy Leather. I can punch through several layers of canvas with the Tandy punches.
It looks like you could pick out the stitches along the top edge and pull the whole lining out. Then you can take the lining apart and use the pieces to make a pattern for the new lining. To do the outside pockets, you'd probably have to take the whole bag apart.
You can use any fabric as interfacing. The non-woven interfacing that you buy in stores is garbage in my opinion, and the woven stuff is too light-weight for most bags.
I like to use canvas or denim. If you want the bag to be stiffer, you can get a stiffer canvas and spray glue it to the back of your shell fabric. Soft canvas or denim used as a sew in interfacing is good for slouchy less structured bags. Just make sure that you pre-wash the fabrics if you plan to wash the bags, or the interfacing will shrink more than the shell and turn the bag into a crumpled ball of fabric.