1) I think you should create any rules you'd like for your sever. I personally don't like the idea of limiting exploration.
2) Again with your server your rules. Again , I think this is a silly requirement.
3)
a) Scavenger hunt would be cool, something with clues (signs) and waypoints and particular items. start and end points.
b) Long boat races : everyone gets their own boat, set a destination etc
c) Cart pull race
d) PVP Colosseum bloodsport
e) Wood cutting, first to get a full cart back to base.

etc etc