if price isnt an issue, i believe turners has a sale until tomorrow on a benelli supernova tactical. that would fit pretty much all the criteria you put forth. thing about benelli is i dont think you can change the stock or the barrel so it's completely single purpose built. but it IS a benelli and it's awful pretty.
You can change out the stock. Once in a blue moon a factory stock will appear on gun broker, but those prices are grossly inflated. Mesa Tactical makes the Urbino pistol-grip tactical stock which reduces the lop to 12". I personally have an Urbino on my Supernova with the limbsaver pad. Nice piece of kit. You can also get side saddles from mesa and rail kits. Nordic makes a rock solid +2 extension too.
I don't know about the Nova (Which is a different shotgun, and I hear it's much less modular than the Supernova), but if you're willing to drop the cash, you *can* exchange the barrels on the Supernova. The barrels themselves are pretty pricey though.

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