I am currently watching it right now and I was wondering, when they beamed Kirk and Sulu back up from falling onto the Vulcan planet, wouldn't they still have died? Since the speed they were falling before they were beamed? Also, wouldn't the gravitational pull from the singularity (Black Hole) be too strong for Spock to walk on the planet, let alone run?
The transporter does not conserve the momentum of the transported, so the speed they were going at is irrelevant.
The relevant part in the script is here
TIGHT ON CHEKOV as he struggles to lock onto them -- BEEP!
CHEKOV: Compensating gravitational pull and...
He HITS A BUTTON and -- 140 EXT. VULCAN SURFACE
5 FEET BEFORE THEY HIT THE GROUND, KIRK AND SULU DEMATERIALIZE!
INT. ENTERPRISE - TRANSPORTER BAY
AND REMATERIALIZE, SLAMMING DOWN ON THE TRANSPORTER PADS, HARD AND PAINFUL, BUT SAFE! Transporter engineers GAPE in utter amazement and relief -- Chekov, sweating, laughs. Kirk and Sulu get their bearings, peeling themselves up, at stunned whispers
So there you go. He compensated for it, presumably in much the same way that the transporter makes allowances for the ship moving at a different speed from the planet below.