Can light emit light? How and why does the Huygens principle really work? I mean, does it always work?
The Huygens principle:

Every point on a wave-front may be considered a source of secondary spherical wavelets which spread out in the forward direction at the speed of light. The new wave-front is the tangential surface to all of these secondary wavelets.

For the Huygens principle to be true, light must emit light. Can light emit light?
 A: Huygens assumed the existence of "ether particles". And every point in wave front, which your statement of Huygens principle is saying to be the source of secondary wavelets, are nothing but the ether particles, which Huygens assumed to give rise to its own individual wave.   
Huygens said that light waves were
longitudinal as they passed through
a substance—the ether. But, now we know that light is tranverse wave because of the work of Erasmus Bertholinus, Etienne Malus, and Augustin-Jean Fresnel on the polarization.     
So, it is the ether particles which Huygens assumed to give rise to wave, and I think this is what you are calling "light to emit light". For your first question: The wave theory which involves Huygens principle can't explain Photoelectric effect.   

Reference: 1. Treatise on Light by Christiaan Huygens (1690) - PDF. Page no.21 (and even the former pages) might be of more use to you. 2. Christiaan Huygens-Wikipedia 3. From Aristotle to Schrödinger - The Curiosity of Physics by Antonis Modinos, Page no. 213.
A: The key phrase is "may be considered".  That is, Huygens principle, as a mathematical procedure, gives (approximately) the right answer.   The principle does not say that light is actually being generated at each point on the wavefront.
Another way to look at it:  the light at a phase front already exists there.  There is no need to generate it all over again.
It's interesting to note that if the vacuum were a "perfect" polarizable material, then the incident light would polarize every point in space, and each point would scatter light into spherical waves.  I don't think that there's any physical justification for the picture, and it might lead to dangerously wrong conclusions, but it might help give you a grasp of how the mathematics works.
A: It's like saying sound produces sound: actually, particles move other particles, and this entire process is sound.
Huygens' principle talks about propagation, not "emission" in the conventional sense. The light is a wave in the electromagnetic field and the changing field at a point in space induces field in the surrounding space when time moves forward. That's the essence of Maxwell's equations. So... do you say that light emits light in order to move forward? Light is a continuous wave phenomenon (let's not get into quantum electrodynamics right now), it changes with time naturally, no need to call it emission.
It is somewhat misleading to say that, bacause light emitting light would mean something else: it would imply self-interaction, self-scattering of photons! Like in two crossing light beams producing light in other directions. That can happen, but only at frequencies so great that it induces particle-antiparticle pair production (~MeV energy is required for electron+positron pair).
When light propagates along the medium, you could say that the response of matter is actually repeated absorption and re-emission of light (polarization, leading to index of refraction and so on). But even in this context, it's unconventional to use this kind of language.
