Would this be at the beginning of game turn or each players turn?
Each players turn.
Does anyone use some other means of determining whether or not a special weapon (or other) has been developed? Or does anyone start off with them in some fashion? If so, how is it working for you?
There are a number of posts in the House Rules section of the forum which deal with Research and Development. Most of those are about how to develop the weapons. Try searching there through those threads.
Thank you :-)