Well, for a RTW trip your itinerary would probably look something like jfk-lhr-sin-syd-lax-jfk which comes in at about 24,000 miles. This would run you 150,000 miles for a business class seat. Also, if you are going to use AA miles for the trip you must fly AA, not BA, across the Atlantic, JFK-LHR.

As for visas, the only one a U.S. passport holder would need to get in advance is for Australia and that can be obtained electronically through a travel agent or the airline.

Sorry I can't help with award availability other than to say the business should be easier than first to obtain.