To answer your questions, yes, this neighborhood does have a community pool and tennis courts. If you want to add a patio in this community, it is likely that you would need to apply to the homeowners association for approval. This home is about 5 1/2 miles from downtown Sarasota, and less than 12 miles from Siesta Key beach.
As has already mentioned, this property was sold nearly 8 months ago. if you are serious about looking for a home, Trulia.com unfortunately is not a very good resource. Your search would be far more productive if you would establish a relationship with a Sarasota Realtor who knows the market well, and who will provide you with accurate real-time information concerning property listings.
There are other active listings in Beekman Place at this time, but be prepared to accept the fact that prices are not the same as they were last fall. You'll want to consult with a Realtor to get an understanding of the current market, so that you can make an appropriate offer and purchase a great home at a fair price.
I will be happy to help you with your search. Please contact me at your convenience.
COLDWELL BANKER RESIDENTIAL REAL ESTATE
423 St. Armands Circle
Sarasota, FL 34236