If it has a mode for it, absolutely not. It is perfectly fine to use the "+" and "-" sport shifting anytime you want. Some models have paddles, some just let you move the shifter over into the gate. But they are built to be driven in either automatic or semi-auto and you will not hurt anything doing it.

Nope, you’re just advising the transmission when to change gears. A transmission operating correctly will not let you choose an incorrect gear or cause any damage.

This feature is useful when deciding from a hill to cancel the engine braking, or to preselect a low gear for a steep climb.

or what I use it for, is to keep a low gear while I’m in rush hour traffic to be able to zip around lanes and have the best possible acceleration.

its more of a gimmick with how smart and automated most new automatics, robotics, dual clutches and CVTs are.

No, it won't hurt it.

What brand transmission?

Or if you don’t know the brand of transmission, what make/model/year of car?

There are some transmissions I’d feel more comfortable playing and having fun with than others. 

Most cars like that have electronics in the transmission and rev limiters on the engine that prevent you from doing something damaging like downshifting at too high a speed, etc.

What make/model are you driving?

Sure go ahead and use it its called an overdrive, if I understood it correctly. Have some fun, or you could use it to get better mpg etc.

"manual shifting" is just an illusion.

It won't allow you to cause damage. That would be a major design flaw if it did. But you will burn more fuel deviating from the program.