Sounds to me like you have dirty pivot points. Your gonna need to loosen the strings and remove the trem springs so you can examine your hardware. Get a magnifyer and look at the fulcrum points on the bride. If there is any "crud", dirt, dust, peeling chrome or gold plateing, you are going to want to clean it up. If the plating is chipped and or peeling, take some 400 grit sand paper and clean off these edges. Also examine the points of contact in the grooves of your pivot posts. They need to be clean as well. Use alchohol and Q-tips to remove dirts and debris.
Also, if the fulcrum points on the bridge are bent of chipped, they will need to be reshaped by using a "half round" jewelers file. The bad part is, if you are allready at this point, it means that your trem is made of a softer metal than a genuine Floyd, and that after tim this problem will arise again. The only permanent solution would be to shell out a few bucks for the real deal, or another high quality licensed floyd.
As far as the buzzing goes, a slack string could be the problem, but I would also check the releif in your neck after tuning down. See, after you tune down, the foward pulling tension on your neck is decreased, and therefore your neck might not have the proper relieif it needs in order for the strings to clear the frets.
Good luck, hope that helps.
