Good choice on a watch, coincidentally I just bought myself a Speedmaster Day/Date yesterday (3523.80). Insofar as the warranty card, yes, get it properly filled out by the merchant. If just they're name is on it though, sometimes that's enough, as you can fill out the rest yourself; date, Serial Number, reference number. The serial number could be tracked to the authorized merchant anyway, so there wouldn't be any problem verifying that.
Insofar as the damage, it may be too late, though I would suggest getting it replaced. Take it back and see what they say, you may be in luck if the watch hasn't been sized and still has the plastic wrapping on the bracelet and caseback. They may say though that you removed it from the store, so they might try to blame you. If it's a reputable shop though, they probably shouldn't. The bottom line is you've got an extremely high quality, expensive timepiece, and one of the first rights is you get to damage it yourself, not somebody else!
Don't worry about the serial numbers or the caseback not having any notches. The speedmaster day/date only has the serial number under the "Speedmaster" Seahorse logo, and there's no notches because I believe it's a snapon caseback (it's only water resistent to 30 meters after all). Even if it is a screwdown though, don't worry about it, I know for a fact that this model comes with no notches on the caseback.
Insofar as the booklet, since it's a gift for someone else, and if neither of you know how to take care of an Omega, I would suggest getting the proper manuals. The merchant should replace them for free, and if not just e-mail Omega or any other authorized merchant and they will send you the manuals for free. I got the manuals for my SMP Chrono that way once.
Here's how you set the day:
To set the numbered day (the outside ring) you need to pull the crown out to the 2nd position (the first position being against the case) and wind counterclockwise. This also sets the month each time it does a full rotation. To set the smaller day calender ("MON", "TUE", etc.), pull the crown out to the number 3 position, and wind the watch as you would set the time. Each 24 hours (or two full rotations) it will move the day. This is also done counterclockwise. You'll know if you're winding in the right direction, as the hands (any of them) should move clockwise as you turn the crown.
