I called it "pendulum step" for lack of a better term. That side-to-side thing is something we used to do back in the 1980s in those Jane Fonda aerobics classes.
I don't know what you mean about jumping rope. Do you mean basically bouncing in place, with the split being the last bounce? If so, I would have more trouble with that, as there isn't as clear a demarcation between the jump rope step and the split as there is between the pendumlum step and split.
See what I mean?
Sorry to be unclear.
I am saying that you should jump rope to train, everyday. Do so just before going on court, after you have performed your warming-up routine. However, if you just warm-up on-court, than jump rope after tennis and before you cool down.