Can you run a BLDC motor backwards without damage?

BLDC motors usually just use permanent magnets on the rotor (be it in-runner or out-runner) and use a set of windings on the stator connected in a three-phase delta or wye configuration. The speed controller just generates a variable-frequency, three phase waveform to power the motor. Since the windings are symmetric, electrically there's no reason you can't turn the motor in either direction.

As for whether it's a good idea to run a prop backwards on landing, that's more of an aeronautics problem than anything inherently electronic. Having flown some r/c planes, it seems to make sense to me that if you reverse the prop on landing, you're basically just applying a braking force along the line of the axis of rotation. If that line passes above the center of gravity (not below it), that should torque things so that the tail will stay down, so you should be stable if that's the case. If the prop axis is below the CG, though, you're looking at forward torques that would drive the nose down, which would result in damage.

Yes, you can drive a brushless DC motor in both directions.

See, for example, the On Semiconductor MC33035 brushless DC motor control chip, which has a pin to control direction.

Here's a little explanation from p. 9 of the datasheet:

The Forward/Reverse input (Pin 3) is used to change the direction of motor rotation by reversing the voltage across the stator winding. When the input changes state, from high to low with a given sensor input code (for example 100), the enabled top and bottom drive outputs with the same alpha designation are exchanged (AT to AB, BT to BB, CT to CB). In effect, the commutation sequence is reversed and the motor changes directional rotation.

I believe you do have to be careful about "shoot-through"-- if you're trying to switch the direction of current flow in a winding, you have to be sure to turn one set of FETs off completely before you turn on the other set, or you may inadvertently short your power supply.

You might google "adaptive gate drive" or "dead time" for more details.

What you have to worry about most when you reverse direction of a motor, is that you do not put too much current into either the motor or the electronics/switches that control it.

When you connect a voltage source across a motor that is at rest and either has a large inertia or a locked rotor, you get a large current flowing through it = V / R where R is the stator winding resistance of the motor. This is called the stall current.

If you are running at full speed with a voltage source across a motor, and you immediately reverse the polarity of the voltage source, you can get up to 2x the stall current, because the voltage source is then at the opposite polarity of the motor's back-emf. This can be too much current, and if that's the case then you have to control the rate at which you reverse voltage across the motor, by using PWM or some other way besides a hard voltage reversal.