Question: There are so many famous paradoxes which are examples of how humans are unable to intuitively understand probability -- there's a discrepancy between their supposed actual experience and the mathematical evidence. There's things like the birthday problem where what we would expect the probability to be is much less than the actual, but also the monty hall problem where the confusion comes in why the answer is what it is. 

My question is, what is the cause of this? Why are we biased into thinking things are more or less likely than they really are? Why do we find it so difficult to accept and understand the correct probability in the case of the monty hall problem, burnt pancake problem, etc.?

Answer: 
As another probabilistic problem consider: What is the probability that all of our intuitions are close to the probability for every mathematical problem? 

In this case the answer my intuition tells me is it isn't 100% and I'm pretty sure in this case it is true.
Most of your questions can be answered by two simpler subquestions:

1. Why does finite combinatorics seems to be so difficult?
2. Why does our intuition is no big help with respect to probabilities?

Even so finite combinatorics indeed contains very "complex" problems, none of the examples you gave involved such "complex" problems. (Not sure in which sense the burnt pancake problem should be related to probability.) Missing experience in judging whether a discrete combinatorics problems allows a simple exact solution, or would better be solved by an approximative method, often leaves us with no solution at all, neither exact nor approximative. For example, is ${15 \choose 5}/{16 \choose 6}$ easy to evaluate exactly? Or should you rather use Stirling's approximation?

If we use our intuition about probabilities instead of addressing the underlying finite combinatorial problem, we often end-up with wrong results. One reason might be that probabilities are combined more multiplicative than additively. And it's also possible that probabilities are really inherently more difficult than finite combinatorics, but we can't believe that. I guess the problem with wrong results arises just from the context in school where we first encounter probability theory. Among all the nice and easily solvable linear problems, we learn this powerful linearization technique, without having much experience with the inherent challenges of the non-linear problems to which it can be applied.

. .3. Why is true randomness so hard to capture, and why can probabilities be hard to get right.

In a certain sense, randomness contains an infinite amount of information, without actually containing any really useful information at all. This is the part of probability that runs directly counter our intuition.