newswatcher
Why did the tail rotor "seizure" not lead to the helicopter "spinning"?
Hollywood movie makers are much to blame for the idea that helicopters spin and drill a hole in the ground if the tail-rotor (or even the engine) fails, but bear in mind virtually every car that crashes in a movie bursts into a ball of flame after a massive explosion even though they rarely do in real life.
If a tail-rotor fails in the cruise, helicopters have enough stability for the pilot to fly straight while he takes the appropriate emergency action to land. The speed of the air flowing past the vertical stabiliser (the tailfin) as the helicopter flies along helps keep the helicopter flying straight.
Think of a weathercock on a church steeple – it always points in the direction the wind's coming from. Wind is a flow of air. Aircraft passing through the air at speed, even on a calm day, create an airflow past the tailfin which causes the same 'weathercock' effect.
Once the helicopter is under control, it can be flown (carefully) to the nearest suitable landing spot. The object is to land as soon as possible but there may not be a suitable/safe area in the immediate vicinity.
The tricky part is the last bit - landing with a failed tail rotor. The passengers were described as 'jittery' - I bet they were - but probably not half as jittery as the pilot because he knew all the things that could go wrong in those last few moments.
For reasons I won't go into, landing without a tail rotor is more difficult in a turbine (jet) engine helicopter (the one in the video is turbine) than in a piston engine. It's also more difficult in this particular helicopter than some because of the position of the various controls you need to use. (The same applies to some other makes/models.) I won't go into detail - let's just say it would be a lot easier if the pilot had three hands!
How fast would it have been going down the runway, at touch-down?
The pilot did a
'run on' landing which is normally about 20 knots (23 mph) at touchdown. That's not fast in car terms but, with no steering, no wheels, no brakes and a risk right up to the moment the helicopter finally stops that it could roll over, I imagine it must seem very fast!
Judging by the video, he seems to be going a lot faster than that at touchdown but you can see from the windsock the runway he's landing on isn't pointing in the same direction as the wind's coming from so he'd have to go faster to allow for that. (All aircraft, including helicopters, land into wind if possible.)
He'd want the forward speed when the helicopter touches down to be as slow as possible but he has to balance that against keeping the speed up enough to stop the nose from swinging to the side. (Remember speed and weathercock effect above.) If the nose isn't pointing in the same direction as the helicopter's moving at the moment it touches down, things can go badly wrong.
The alternative is an
'engine off' autorotation with zero groundspeed at the moment the helicopter touches down. The pilot starts a descent, closes the throttle(s) - again contrary to Hollywood image, helicopters can glide although not as far as aeroplanes - and lands. The touchdown may be a little 'firmer' than normal but experienced pilots can often do an 'engine off' as gently as a normal landing.
Which method is adopted ('run on' or 'engine off')will depend upon several factors, including the manufacturers' recommendation - their test pilots will have determined which is better for that helicopter type.
It may all seem easy but, as you've probably guessed, it's not quite so easy in practice. The Las Vegas pilot coped with the problem superbly, without anyone being killed or even injured, and he did it at night!
We're all trained to cope with a loss of tail rotor, and are regularly checked on our ability to do so, but I've never had a real failure and I'll be more than happy if it stays that way.
NB: The above is intended only as a simple summary.
Also, I'm not an instructor so apologies if my answers aren't as clear to you when you read them as they seem to me as I write them. 
[Edit]
Like the questioner, I assumed the tail rotor was seized. If the pedals were jammed (see Whatsarunway's post below) and the pilot had no control of the t/r, then my response would have been different in a number of respects but, given the general nature of the questions asked, I'll leave my answers as they stand.