What is the term take- off?? Also the term Fixture carrier and fixture rough??

I am not sure what you are referring to in the way of plumbing talk but in the entire building industry take-off means to read the blue prints and make a material list of what is going to be needed to do the job, it means other things but that is the main thing.

The only fixture carrier I can think of is a support that you put in the wall to support a commode or a lavatory, the carrier for a wall hung commode is also a part of the plumbing.

Fixture rough, I think would refer to the way that you would lay out and install you pipes for the (rough-in) portion of the job, next would come the Top-out and then the would be the set-out.