Can a taildragger land tail first?
It's possible to land by touching the tail first, but this is undesirable as it puts a lot of stress on the tail, so it's not trained for and there are no circumstances where you would do this deliberately. There are two common methods of landing a tail-dragger: a three-point landing (where the tail and mainwheels touch together) and a wheel landing (where the mainwheels touch first).

Aircraft design

First, be aware that the aircraft is not level when it's on the ground, like a tricycle aircraft is. The tail position is set so that the aircraft is quite nose-up on the ground - in fact, the angle is about the same as the stalling angle, and the reason for this will become apparent. If your nose is on the horizon, the tail is off the ground (about five feet up in a Tiger Moth). Also, the prop is mounted higher up on a tail-dragger than on a tricycle, for exactly the reason you say. In the Tiger Moth, there is two or three feet of clearance between the ground and the prop if the nose is on the horizon.

The clearance itself isn't the direct problem. If you do go nose-down on the ground, the airplane is not stable. If you're moving forwards, the chief component of drag is acting through the mainwheels, so the engine thrust will tend to tip you further nose-forward (especially if you run over a hump or rabbit hole). In addition, the CG of the aircraft is closer to the mainwheels, and eventually goes in front of the mainwheels. If this happens, a prop strike is inevitable.

Three-point landings

Three-point landings are the default because the aircraft is fully controllable as soon as it touches down. The approach to the runway is done at low or idle power, and set you up to be flying level at rooftop height above the runway. As you lose speed (because you have idle power), you progressively pitch up to maintain height, which in turn causes you to lose more speed. This is called the flare Soon enough you are pitched up at the three-point attitude, and you can no longer maintain height - if you pitch up any further the wings will stall. At this point, you will drop down onto the ground and all three points will touch at once.

Wheel landings

Wheel landings are also used, as they offer more control in windy conditions, but they typically have a longer landing roll because you have more speed when you touch down. The approach is flown with some power on, but less than cruise power. It's a gradual descent towards the runway, and there's no flare, so you touch down in the level attitude. You then close the throttle to idle, and let the speed decay on the ground. As your speed decays and less lift is produced, you need to progressively push the stick forward to keep the tail from dropping, keeping the aircraft level as long as possible. Once the tail starts to drop, pull the stick back to drop the tail firmly onto the ground.

When do tail-first landings happen?

To reiterate, you never land tail-first deliberately. It offers very little control; the aircraft would stall at a higher speed (and therefore the landing roll is harder and longer); it puts undesirable load on the tail. They do sometimes happen by accident, if a student pilot has pitched up too aggressively when trying to do a three-point landing. Typically, too much nose-up input results in "ballooning" as the aircraft gains a little more height, but this results in the speed decreasing rapidly. If left uncorrected, the aircraft stalls, resulting in the tail touching down hard.