The kit used for military parachuting is much different then the civilian gear you might use. There are jumpmasters and parachute instructors on this site who can add more or correct my info:
For a basic parachutist, the gear is a CT-1 conical chute activated by a 24 ft static line. The reserve sits on your chest and is activated by a pull handle. The drills do not involve cutting away anything, but are taught in a way that minimizes the risk of fouling your chutes.
There is also a CT-2 chute which is somewhat steerable. It has a forward direction of about 8 knots and is steered by rotating on it center axis. But is is a conical chute as well.
The free fall chute, which is very similar to civilian chutes is the CT-6. it is rectangular and is steered and flown in the same manner as you would a civilian chute.
For static line jumps, the minimum safe altitude is 800 ft above ground level. Most training jumps take place at either 1000 ft agl for trained jumpers and 1250 ft agl for trainees. The absolute minimum altitude is 400 ft.
Talking with US soldiers who jumped into Panama say at that altitude you might have time to look up at your chute, and then you hit the ground before you look down.