Replacing a ceiling fan, how to connect the ceiling mounting bracket to the existing electrical box , there are two tabs? On the electrical box in opposite corners, are these used to connect the mounting bracket, if so that will rotate the mounting bracket to an undesirable position.

I am not sure of there is a question in there somewhere. The way it is written I can't tell.

You should be able to mount the bracket to the box and hang the fan in the bracket.

Is this a hanging ball style mount, or is it some form of rigid mount?

The mounting bracket usually has two oval shaped holes that line up with the threaded screw holes of the ceiling box. You use the appropriate size screws (included with either the fan or the ceiling box) to mount the bracket to the box.