Does aloe vera clog my pores?

No, pure aloe vera (e.g gel 98%) will not clog your pores. You should aim for 96% + aloe vera gels if you want it pure. Better yet, an aloe vera plant.

What does aloe vera do for my skin? It can enhance scar recovery, you know those damn red spots.. It can calm your skin (it can aggravate if you react on it) and reduce redness. I also use it on, sorry for the direct language, open zits. You know the ones that seems like they will never stop pushing out liquids? Put a thick dot of aloe vera on the spot and wait 30min before you apply your moisturiser. It will make sure the yukky liquids don't run down your cheek in worst case scenarios.
I do not recommend using aloe vera alone on the entire face, this can be very drying for some. Apply a very thin layer and let it soak in, then apply a moisturiser.
The thicker the better? No. Thick layers will be much more drying then thin layers. Test it yourself if you want to.

If aloe vera makes you break out, it's because your skin reacts negatively on it.