this is a much better answer than mine! ♥️
hukugame
with flash photography, shutter speed doesn't matter that much. If you want to use shutter speed faster than 1/200, or 1/250 for most cameras, make sure you have high speed sync on. your flash will pretty much fire at the same speed all the time (about 1/1000th of a second), and high speed sync just means your flash will try to sync up with your camera's shutter speed, by using very fast burst of light, so you don't end up with black lines on your shot.
so regardless of what shutter speed you use, your subject will be lit and frozen, and like you said, using slower shutter speed will introduce more ambient light into your scene, and may even create a cool blur around your subject.
when you use flash, only thing that really impacts your exposure a lot is your aperture.
Magic Meghan Photography