Should you manually put the system in storm mod every time it rains

I don't know what you call storm mode.   don't know where you are located but usually in Canada and USA there is rules about overflowing water into water stream so your system should work as usual, pumping water where it should go, meaning to the wastewater plant.

It really depends on the system. Our local plant has done a fantastic of getting the storm water system separated from the sanitary system.