Sean, my favorite place growing up did 21" slice pies. If I had the equipment to handle 21" pizzas I think my heat would explode I'd be so happy.
There's also the places in Hoboken that do slices as big as your arm. It's been almost 20 years since I've had a Hoboken slice, and although the crust and sauce were not as good as my local place, the size was magical.
Bigger slices of pizza are almost always better. This is always why I push people to get the largest stone that their oven can handle, with a correspondingly large peel.
I'm sure that one of the New Haven folks will chime in extolling the virtues of oblong pizzas, but, for NY, you really should strive for round. If, in your attempts for a round pizza, you end up with something oblong, there's nothing to be ashamed of, but purposely setting out to make an oblong pizza because that's the size of your stone, no, not for NY.
As far as extending the skin beyond the peel... the peel already has sloping edges, so if you go all the way to the edge (or past) and have a small rim (recommended) with sauce near the rim... the sauce is going to slide.
I say work with what you've got. 18", round and don't hang off the edge.