Interactive 3D graphics require webgl. Webgl is disabled by default in Safari on Mac. To enable it do the following:

In Safari, open the Safari menu and select Preferences. Then, click the Advanced tab in the Preferences window. Then, at the bottom of the window, check the Show Develop menu in menu bar checkbox. Then, open the Develop menu in the menu bar and select Enable WebGL.

That being said, the Cloud implementation is not as fully featured as the desktop. In some of these cases we return a static image instead of interactive content. We are in the process of adding better messaging to indicate when webgl is disabled or not supported by your browser and when the specific 3D graphic is not fully supported in the Cloud.

Correct, 3D Graphics within a Manipulate are not currently interactive. It is something that we plan to have in the future.

Safari is supposed to be compatible.

I tried just now with Safari 6.1.4 and Firefox 30.0 on MacOS 10.8.5.
In Firefox I could rotate the graphic but in Safari it was inert.

In case it makes a difference, my example input was

Plot3D[ Sin[x*y],{x,0,3},{y,0,4}, ColorFunction->"Rainbow"]

I will report it to cloud developers. I should not predict how long to get a correction in place.