Dragon will land on Earth using only the propulsion from its rocket engines. Has any operational or test spacecraft done that before? If not, what is the highest altitude from which propulsion-only landing has been demonstrated?
I think DC-XA in 1996 has the record, a landing from 3km altitude.
More recently, SpaceX's Grasshopper and Falcon development vehicles have done powered landings from altitudes on the order of 1km, and a number of other companies such as Armadillo Aerospace and Masten Space Systems have demonstrated powered landings with small test craft from low altitude.
More info here.
The Apollo Lunar Module did propulsive-only landings on the Moon from an altitude of 110 km and a speed of 1625 m/s.
For Earth, propulsive-only landing from orbit is not popular because the weight of the fuel would be much higher than the weight of aerodynamic landing equipment (shielding, parachutes).
Also, even in a propulsive landing, atmospheric drag is going to contribute something to the deceleration. Pure propulsive landings on Earth would be very inefficient (decelerate to 0 above the atmosphere, then keep running the engines to slowly descend through the atmosphere).
"can" and "should" need to be applied here. Rocket-only landings on a large planet with a substantial atmosphere are a very poor choice. Parachutes are MUCH lighter (although somewhat less accurate), wings are very accurate assuming you have a place to roll out your forward speed.
Skimming the Draco page is a bit confusing - are they using rockets all the way down, or just at the end? A Soyuz-type landing is basically a parachute descent - the 2 seconds of rocket thrust at the end are solely to make the impact less ainful for the passengers.
Note also that the Space Transportation System meets the requirements set by the OP:
no use of parachutes (and no bouncing airbags). Heat shield and chock absorbing landing legs, and maybe before that aerobreaking, might be practically complementary.
The Orbiter would use the OMS to propulsively manoeuvre out of LEO into the atmosphere, and then aerobrake until touchdown. Note that Endeavour was in fact fitted with a drag chute that would deploy after touchdown, and drag chutes were subsequently retrofitted to most (all?) other orbiters. However, the orbiters were capable of, and performed many, landings without the drag chute.
The Dragon V2 is designed to use areobrakeing first to slow down then rockets to land at the end without parachutes, however, they are retaining the parachutes as a backup. SpaceX is going for rapid reuse-ability not total efficiency since it is much cheaper to do 2 200 thousand dollar flights then 1 100 million flight.