A quarter sheet pan is tiny ... 9 x 13. You'd have problems fitting a large chicken on that. If you don't have a place to buy a half-sheet pan, look for places that do appliance repair, and get a broiler pan ... they're of a similar size, and have plenty of uses. (they just don't work for cookies)
In the comments of a Serious Eats post on spatchcocked turkeys was this exchange:
ColoradoTommy ; 12:59PM ON 11/06/12 :
Kenji, what is the maximum size bird you would recommend spatchcocking (sp?), before going with the two bird approach?
J. Kenji López-Alt ; 1:01PM ON 11/06/12 :
Unless you have an oven large enough to fit a full-sized sheet pan, you won't be able to do more than a 12 to 13 pounder, max—it'll overhang the edges.
Even with a bigger bird, you'll have a few problems since it takes longer to cook. You'd have to reduce the oven temp by 25 to 50°F, and allow it to cook longer. Just use a thermometer, keep an eye on it, and you should be fine.
I'm not sure if his recommendation is for a 1/2 sheet (13"x18"), or a 3/4 sheet (15"x21") ... I know it's not for a 1/4 sheet.
As for cutting out the backbone -- shears would be the standard way, but for chicken I've often just gone in with a sharp knife and slowly cut through everything. It doesn't slice through easily, but if you keep one hand on the backbone to keep the chicken from moving, you can slowly saw your way through the ribs. (mind you, I've never tried on a turkey).
If you do get sheers, I like the ones that separate to clean and that have a little notch near the pivot to make sure bones don't slip as you cut them. The ones I have were sold as 'kitchen shears' not 'poultry sheers' (which are spring loaded (ie, more places to worry about gunks), and tend to be $70+, while my kitchen sheers were closer to $20).