Let's hear it for science!! Or at least for logic, since it seems that these days science is at times becoming a belief system instead of the process it is supposed to be. :)
What your teacher from Foxwood told you about fae existing in another dimension is basically the same sort of thing Mr. E, the elf Susan channels for this forum, has said. (Technically I'm not sure 'dimension' is the correct word, but it's probably the best thing we've got right now.) My own impression is that they are sort of a jump up and a nudge to the right of us…or at least, definitely a jump up to a vibration that is faster, and translates to us as thinner/less dense than the vibrations of this physical plane we all know and love. I suspect the 'nudge to the right' bit on account of how Mr. E also said that it's possible for them to simply not perceive things from our world, whether those things are living or not. (Humans, for example.) Either way, the worlds are intimately connected; Mr. E has also mentioned that the geography of his world/plane/dimension is very similar to the geography of ours.
The explanation your teacher gave as to why they might appear differently to people is interesting. Of course, don't we all tend to view the world through the filter of our own psychologies anyway? It's possible that what's happening is that the less stable perceptual information we get (like visual, audio, etc.) about the fae, the more work our brain has to do to fill in the gaps using our own preconceptions. Or again, since the fae are said to be able to shapeshift, maybe they really do shift around a bit sometimes just to mess with us. :)
Time…well, that's a tricky question I've been pondering for a long, er, time. I know Einstein said time was the forth dimension, but I think it's a trick question. It seems to me that if you think of the 4th dimension as a cut gemstone, then Time is just one single face of it that we happen to be able to see according to the way we currently exist. My current thought is that Time is only a true concept for a given value of "true". Scientific experiments have already proven that time is not a constant, and if you try experimenting with it a bit, you may find some evidence as I have that time is related to your frame of mind. (This may possibly also be true for space, which would argue for one's experience of the entire space-time continuum being a function of one's mind, as well as raising the question of just exactly what the space-time continuum really is.) Now, Mr. E has already commented that time passes differently in his worldframe, so this raises some very interesting questions about the nature of how the two worlds (ours and his) interrelate.
As for fae being untrustworthy, I think that's just the problem you get when two cultures are unfamiliar with each other and don't know really what to expect from each other. Each culture has some unspoken assumptions and rules that may not even occur to the other culture, so there's lots of room for misunderstanding and hurt feelings on both sides if people can't kick their egos out of the driver's seat, so to speak.
But all of the above simply explores how fae and humans relate to each other. To ask 'what are the fae'…well, you know, imagine going into the Star Trek universe and asking 'what are aliens?', and they will ask you "Which ones? Vulcans, Romulans, Klingons, Andorians, Beta-zoids, Cardassians, Borg…? Which ones did you want to ask to tea?" With fae I think it's much the same sort of thing. Once you get past saying 'the place where they live is just a jump to the left and a step to the right of us', then you start getting into the really interesting and complicated areas. Elves, Dragons, Leprechauns, Pixies, Fauns, Stickmen…which ones did you want to ask to tea? :D
Anyway, these are just my thoughts and conjectures. You've raised some really interesting points and questions here, and I do hope others will chime in!