Anchoring, from our experience:

Norman - unlikely you would need to but you can always anchor out the back, plenty of room, frequently see others doing this (generally larger boats). When on our own yacht we always did this.
Peter - Great Harbour - Not huge amount of room where depth and space suits anchoring, but enough for a handful of boats.
Peter - Little Harbour - yes, but take a stern line ashore.
Cooper - only really if you're a BIG boat with oodles of chain, in which case yes but well outside of the area where the mooring balls are. We've had no issues on 70ft+ boats but wouldn't attempt with the average charter boat. Or in right conditions round the corner in Haulovers Bay.
Marina Cay - yes, plenty of space to anchor, again outside of the mooring-ball area. Often also seen smaller boats anchoring between the reef and the mooring-ball area. Its not for me...but each to there own.
North Sound - Vixen Point and round into Prickly Bay, great anchoring. Or just out the back of the BEYC mooring field...you'll have the place to yourself. Lots of other spots in North Sound also.
Anegada - yes, per crusing guide and also Pomato Point.
JVD - Diamond Cay.
JVD - Great Harbour, not as much room as there used to be and if prudent you won't be tucked in very far, but still a perfectly good option.

Generally always a decent anchoring option if you find balls all taken. With the exception perhaps of Cooper.