Not something I have any experience with but please allow me opine from my armchair:
the only problems I forsee with that approach are:
-- any bends you might have to navigate
-- supporting/stabilizing the new pipe
-- sealing the top to prevent a down draft forming between them and pulling exhaust into your home
There are no bends, or I wouldn't even consider it, and figuring out the support/stabilizing of the new pipe would likely tie into sealing it to prevent the downdraft.
Yes, but I need to make sure it doesn't increase the airflow by a very large margin, or it affects the stove in a very negative way (either burning dangerously hot, or causing smoke to go the wrong way, etc).
Mostly I need to inspect what's currently in place and see what fits where to figure out what I am doing.
My sister's creosote build up in her exhaust pipe ignited one Thanksgiving. A fire of sticky tar, in a tube running through inaccessible walls and roof. That was interesting (ripped the pipe out quickly and it was contained). You may want to inspect before adding in a new exhaust, if you haven't yet.
Chimney fires are incredibly scary, and I definitely will be cleaning the old piping before I do anything else. Fortunately I don't have any sort of attic or complicated setup, it just goes through about 1' of ceiling/roof and that's it.