Lots too answer, here.
I am in agreement with Mirage about our subjective belief that we can "control" things. I know how often the unlooked-for side-effect smashes all our efforts. But we grasp and we try, and we get better at it, little by little. Is it an upward climb by humanity towards god-hood? Who knows, but the writers of Star Trek, The Next Generation frequently think so. And so, possibly, do I.
Baruch, I'm actually a very emotional person. I respond viscerally to a great many things, with tears, a lump in the throat that makes swallowing difficult, with great joy or laughter, with anger sometimes. I am a huge lover of Shakespeare (I see myself as sometimes Romeo -- with a Romeo partner

-- and sometimes as Benvolio. I am frequently a peacemaker). I'm actually a huge lover of theatre of all kinds. In fact, tonight I'm going to see a pair of plays, Black Comedy by Peter Shaffer (a farce in which the audience can see the light but the actors can't) and The Real Inspector Hound by Tom Stoppard, another farce which skewers the murder mystery genre. Then on Sunday I'll be going to the Stratford Festival (one of Canada's best Shakespearean etc. festivals) to see "The Trojan Women" by Euripides, which will probably be my last chance to see one of our greatest actresses, Martha Henry. Then two weeks tonight, it'll be Ring Round the Moon, a translation of "L'Invitation au chateau" by Jean Anouilh. Joseph and I see at minimum a dozen plays, musicals, operas etc. every year. It gets expensive, but it's our one major indulgence.
Back on the subject of emotion, I have been known to actually cry during a performance. After the play "Night, Mother" by Marsha Norman earlier this year, I couldn't get out of my seat for several minutes. A little embarassing, but I respond to all emotional stimuli with emotions of my own. Always have. But I think, too. I've actually been measured with both a pretty high IQ and EQ (Reuven Bar-On's test). I don't put all that much stock in either.
Part of what I think might be at least slightly "spiritual" about me is that I recognize that I cannot understand everything. I wrote the following a couple of months ago, and repost it here for convenience:
I think we conflate spirituality with religiosity, and yet they are not the same thing. It is quite possible to be religious without having much of a spiritual side to you at all, and yet it is also possible to be an atheist, and remain quite spiritual.
Consider, for a moment, Buddha. Almost everybody thinks that Buddha was a very spiritual person, and yet he never expressed a belief in any god, and even urged his followers not to spend any time speculating about such things, but rather focus on this world.
Spirituality is what happens when we contemplate those things for which we cannot find answers. We are a finite species confronted with a universe that is infinite, so far as we know. We are a very temporal species (quite short, actually) in a timeless reality.
These things are deeply mysterious. We cannot make any real intellectual sense out of them. Therefore, we have to probe ever more deeply into our very selves, our understanding of what it means to be us, finite in an infinite universe, perishable in a reality that will not perish. What can it mean? Asking those questions, and looking inside of ourselves, is what spirituality is.
The moment we realize that we know a great many facts, and can perceive a great amount of the reality of the world around us -- and at the same time realize that we don't, not really, know what they are for -- and then begin to wonder about the answers to that simple question -- "what are they for?" -- we are spiritual.
But many people imagine that, in a frenzy of over-wrought emotionalism, probably brought on by too much dancing, too many shouts of "hallelujah," or possibly the after-effects of a misspent evening, that they have experienced "a rebirth," some sort of transformation caused by the acceptance of this saviour or that into their lives.
This is emotion. Our emotions, based as they are on what is uniquely our own and nobody else's, which cannot be felt by anyone else except through empathy, are one form of spirituality. Our ability to love (or to hate) are spiritual because they speak to our deepest understanding of ourselves and those for whom we feel those emotions.
It does not require a belief in god to be spiritual, for which I am grateful. After all, there is simply no evidence that there is such a being, and yet I still wonder about so many things. If I couldn't be spiritual because I didn't believe in god, I would be terribly diminished.
And yet, I recognize the mystery of life. I am constantly amazed that the more I learn, the more mysterious it all gets (that's what Socrates meant), and that the ultimate answers get pushed further and further away. For example, when I was a child, I learned that everything was made up of atoms, too small to see, and even that atoms were made up of even smaller bits, called electrons, protons and neutrons. I was happy. I "knew" how everything was made, and a little study of the periodic table helped me understand how it all fit together to make a real world. Then Quantum Physics came along. Then String Theory (which I understand not a tiddle of), and all my truths -- for all my study and learning -- got pushed further and further away. It turns out I know less, much less, than I thought I did.
But my endless longing to know, to understand, to figure out how it works -- and how I am a part of it all -- that is spirituality. That doesn't require god. It requires me and the universe, that is all.
I could as easily suppose there was a god there as not, and yet I do not because I see no evidence to support such a supposition. Making things up, inventing answers, does not seem to me a good way of getting at the truth of things. And for that reason, I also think that making up gods to answer the unanswered questions is not spiritual but is, rather, a step away from spirituality.
That is not to say that a religious believer cannot be spiritual, either. Of course they can. But they are spiritual in the same way that I am spiritual -- through the endless wondering about themselves, the universe, what it all means, where it all came from, where it is all going.
As an atheist, I do not feel the compulsion -- which I think that some religious people feel -- to pretend to understand all of the mysteries. I don't need to create answers (and label them God) to explain everything that I cannot understand.
Now, as to the other points that you raise, I have to point out -- as I do so very often on this board -- that the business of "spirituality" that you describe as "experiences," this business of coming into contact with "higher powers" is, in my view, much more likely to be nothing more than tapping into your own subconscious resources. Whether you do that through meditation, through frenzied dancing or handling dangerous snakes, or taking drugs, or standing on your head for 10 minutes and then standing upright so all the blood rushes out and you get dizzy, or by any other means is not important. The important thing to recognize is that, until somebody gives me a reason to believe otherwise, I will always believe that these "experiences," this contact with something higher or other, is really generated inside of our immensely complex brains. They are as much an intimate part of us as being able to see the blue sky.
Ultimately, I believe that trying to understand the universe, and our place in it, by forcing religious explanations and dogma onto the unknown is to miss the point altogether. You cannot learn or understand anything new when you are no longer asking the questions, and that is what religion essentially asks us to do -- accept the religious answer as "the truth," and stop asking questions.
You are much better off using simple "spiritual" means to seek understanding -- means like intuition, your emotions, wonder, and ultimately to accept that if some things are ever to be really understood, that understanding will come with time and real evidence. But if the answers don't come, spirituality means also being able to accept that, and continue to wonder.