In Cat 1 approach, the DA is based on a barometric altitude. Look at the approach plates, the DA is runway elevation plus 200'. this equates to approx 200' from the ground, taking into account the terrain in the approach it may be higher or lower than 200' on rad alt but never lower than 200' from the runway.
It is an alt that at which you decide to continue for a landing or go around to execute a missed approach. There fore you can desend below, as you say for reaction time etc,,, for example heavies in a cat II approach allow for the wheels to touch the runway, but by this stage you have set G/A power, just the mommentum taking you below.
ILS approaches are precison approaches so the DA reflects this and allows you to desend below, not to sneak below for a better look, just the time it takes to pitch up and set power etc.
VOR or NDB approaches are non precision, therefore have the MDA and it is a hard deck, do not go below. Many operators will set the descision call 50' above the mda, this means you can perform a constant rate of desent method, ie pretend its an ils, and work out your heights (in advance) from the threshold, this works better than dragging the A/C in along the MDA then deciding at the MAP cutting the juice and dropping on the runway. You decide 50 above and can execute a GA without fear of breaking minimums...
the two crew works with PNF calling the heights, IE one hundred to go , or SOP equivialant, then decide. the idea being that he/she is looking in at speed, alt, engine parameters etc. But the roles can be reversed in Low Vis Proceedures, whereby the PF's head is looking in, PNF decides at minimums, calling either contact or G/A.... dependig on SOP he may continue for landing by taking control or calling g/a in which case the PF will execute,