The duration of the flash is much shorter than 1/30 sec., and in most indoor situations, the contribution to the exposure from the flash will be at least four or five stops more than ambient, so the flash will stop any motion. The effective exposure time with flash might be anywhere from 1/24000 sec. for a small shoe-mount flash at very low power to 1/300 sec. for a very powerful studio strobe unit. You only need to worry about blurring or ghosting if the ambient light level is very high relative to the flash (say if you were combining hot lights and strobes or bright daylight and strobes), or if the shutter speed is very slow, say longer than 1/8 sec.

You can use slow sync speeds indoors to take advantage of the ambient light, but then you need to be careful about camera and subject movement (though sometimes the sharp subject with movement around it can also be interesting).

The modern strobes that behave like FP bulbs for high-speed sync on a 35mm SLR are usually dedicated to the camera and won't work that way with non-dedicated systems. They also have the disadvantage that they can't output full power in high-speed sync mode, because they can't recycle fast enough to flash repeatedly at full power in the time it takes the slit to cross the film gate.