I personally think that there is a limit on n for which nth row templates exist (one stone connecting to the edge) even if the side they are connecting to is infinitely long. However, I have no theoretical basis for this - just a gut feeling.
A couple years back I tried and failed to prove this... had trouble with defining the end of the game when the template could go (infinitely) above the nth row. Kind of reminded me of Conway's Angel problem, where an Angel win is an infinite game :) Anyways, the way you guys have defined nth row template (just use empty cells up to nth row) sounds more provable to me... although still far from easy.
If anyone is still interested in this, I would suggest that exploring 7th row templates might best be done in a contrary style: start by proving that there is no template in a 7x7 space, then in a 7x8 space, then 7x9 and so on. That should be doable, perhaps even by an algorithm. After a while the spaces will either find a template, or find the spaces are so broad it doesn't matter in any practical game size. I don't know if anyone is playing bigger than 19x19 at the moment, but go as high as you like.
You could also make it a bit more interesting, by looking for 2-stone, then 3-stone (etc.) templates at each level. I've got my own stuff I'm working on in my doddering old age, so I'll probably never get around to it.
Little Golem