Well, I must say that your apprehension makes more sense now. Thanks for sharing the information that we need to help you out!
I agree with what's already been said. Just tell them what happened, because it will more than likely get found sooner or later. The security clearance review is no joke, and if you stay in for a while and get an even higher clearance they dig deeper.
The drug involvement is a big deal with the Navy (and the military in general), and for good reason. We don't want people working on equipment under the influence of drugs or alcohol. You can even be removed from the program for getting a certain prescriptions! Own up to it... You'll have to do a lot of explaining, they're going to want to know every detail. The longer ago the event was, the better. This will prove that you've straightened yourself out. You will still need to have a waiver to get in if you admit to drug use. There's a lot of people with those waivers is the Navy too, so not a huge deal.
As far as why you were dismissed for school, I don't think the Navy is going to have a huge issue with it. You weren't convicted of anything, and if I remember correctly that's all that the enlistment paperwork cares about. Not saying that they'll definitely not care, but there's no legal record of those things happening.
Going back to earlier about the recruiters. Don't let them convince you to lie! I can see in this situation why they might try to do just that. Remember that you're the one signing all of the documentation, and read the little words that are around that signature line. They can legally punish you for knowingly providing false information! If you think for an instant that some shady recruiter is later going to admit that he pressured you to lie you're way off!
I hope things work out for you. Some final words of advice if this all does work out and you get into this program... People make some silly choices when they're young (and older, trust me). Learn from the stupid stuff you've done, and remember the Navy will have no tolerance for any of that stuff that got you kicked out of school before either. Biggest takeaway from all of it... change the way you're thinking, and if necessary the people you're hanging out with. They would be no help to you while you're in the Navy, just a bad influence. This is an opportunity to better yourself, take advantage of it!