I've been making peanut butter cookies, but skipping the step of pressing the top of the cookie down with a fork. What does this step do for the cookies?
Peanut butter cookies don't spread as they cook, so you have to flatten them before hand. This ensures that the middle will cook through before the outside burns.
As for the pattern created, it actually creates slightly more surface area, so you'll get more browning at the extra edges that you create. Think of it like a meringue, or the top of a shepherd's pie -- if it's too smooth, you won't get the little bits of browned crispy bits that you'd get if you rough up the surface.
I believe the main reason is to help the dough spread out. Peanut butter cookies recipes typically don't go for a lot of spread, and therefore you need to manually flatten the dough in order to get a proper cookie shape and allow the cookie to cook properly.
It's not clear to me whether the non-spreading recipe is a requirement for some aspect of the cookie, or if the reason that a non-spreading recipe is used is specifically to allow the fork-marks to remain, so you could experiment with some of the factors mentioned in this thread on cookie spreading if you'd prefer the cookies to spread on their own and skip the fork marks.
I actually just asked my wife about this! It turns out that the dough used for most peanut butter cookies is a little thicker than regular cookie dough. Pressing it can help it to cook more evenly. I don't know how much it helps, but she seemed to think it was pretty important!
The fork lines are so you can tell the difference between the peanut butter ones and the sugar cookies.
Nothing. It's just cosmetic. I didn't even know I was "supposed" to be doing that until I'd been making them for years.