I like to think of probabilities as possibility trees. Here's an example of one such tree. Also, probability is entirely easier when one remembers that the total probability of events is 1; that is, the probability of a specific event e not happening is 1-p(e), where p(e) is the probability of e happening. Also, dice are independent measures, so what one die rolls does not affect the others.
For example, the probability of no haunt happening on the first roll is 1 - probability_haunt. This is much easier to calculate than totaling all combinations of probabilities. A haunt happens only if all dice roll omen (1/3)(1/3)...*(1/3) = (1/3){12} = 0.000188167% of the time.
While it's possible to calculate probabilities successively building up from what we know about omens at the first check, our probability tree would quickly get rather confusing. This is precisely why statisticians study Binomial Distributions. That is, the study of probability when we have two options (Yes/No type questions (Omen/No Omen)). The crux here is that P(X<=x) = P(0) + P(1) + ... + P(x). Using this, we can quickly apply it to your specific question.
The probability of rolling a haunt at the 6th omen is P(X <= 6) (that is, we roll 6 or fewer successes on our dice). The formula above gives that this should be:
(1/3)12 + (12 choose 1)(2/3)(1/3)11 + (12 choose 2)(2/3)2 (1/3)10 + ... + (12 choose 6)(2/3)6 (1/3)6
Google and Wolfram Alpha say this should be roughly 17.8% chance, which sounds about right to me, intuitively. When you're looking at passing 7 dice though, it immediately jumps up to 36.8% chance of failure. By the time you're staring down the dice, looking to pass your 10th omen, you're looking at slim odds of success: 94.6% of the time you'll be going into the haunt.
Note: this calculation only deals with probabilities given that you're already passed the previous tests. That is, we've not calculated the probability that *the haunt will start by the time you've had to pass 6 omen checks). That additional step is relatively easy, and is left as an exercise to the reader[s].
