Depending on the exact management position and the exact types of supporting experience, perhaps. In general, the experience is the most important factor in getting a management position, the right degree helps but the wrong degree surely hurts.
It might, provided you manage to crowbar some engineering experience in there. Doctorates are very, very tricky when you want to step right into management.
There are substantial differences, but SE is more about expanding a core engineering principle (usually EE or ME) to a higher level, while IE is a more general field that is less dependent on prior experience. Given that you lack a lower-level engineering degree, I think IE is probably a better choice for you.