Welcome Axell! I'm also new to Pathfinder, but have been playing D&D for nearly 30 years. The Beginner Box is GREAT!
1) As I like the idea of keeping my game BB only, I bought the Bestiary for new monsters, but haven't yet purchased the Core Rules. The stuff I do use from it is free and legally online at http://www.d20pfsrd.com/ so I can pick and choose the elements I want (such as levels 6-8). Either way, those two books are a great way to continue OR supplement your boxed set adventures.
Definitely check out the free BB downloads here on paizo.com too. They have more of everything imaginable for the BB and are a great source of additional fun!
2) I use the blank map from the box with wet-erase markers, and erase as needed for space. The flip-mat basic you buy (not the one in the box) is double-sided blank, so you could always use both sides for your two level dungeon that way.
3) Precisely what IejirIsk said: Yes. Always roll with it if the players are clever. Enjoyability, humor, and pacing always trump tracking down specific rules IMO.
4) If you had a blast... and it sounds like you did... just keep on keepin' on! Way to go!