White domestic geese do usually have orange beaks. There are lots of breeds, but most in Britain are just "farmyard" mixtures.
Geese are hard to sex, even though waterfowl are amongst the few birds in which the males have a penis at all. When not in use the penis is well tucked away, and you have to get quite personal to get it to protrude.
She's laying at this time probably because she's young and just starting (pullets do the same). In future years she'll probably just lay for a while in spring, unless you mess about with her day-length. (All assuming you're in the northern hemisphere...)