1) Is there a set of "rules" that the universe MUST follow?
This is a nice, broadly phrased question to get the conversation going, but I think we should focus it a bit more. The way this question is reading, it looks more like it views natural laws as something normative rather than something descriptive. But this isn't the case.
Any laws that we observe in the universe are descriptive. They describe how the universe works rather than imposes some sort of restriction on how it can or ought to work. The natural laws that we observe must rest on some big assumptions:
1) that the universe is pretty much how we perceive it to be
2) that these descriptive laws are actually the case, i.e., we're right about physical laws and the theories within
3) we have exhausted all the natural laws within the universe
1 is fairly uncontentious, although there are a few skeptics running around who would argue 1 is false. But that's missing the point completely and this kind of argument has no place in our discussion
2 is proving more and more difficult to demonstrate. We thought we totally had it with Newton's laws, which were replaced by Einsteinian physics, which was itself replaced by quantum physics. If history is any guide, then we likely still haven't gotten it right. Then again, we would eventually be able to get it right at some point - I would hope.
It's 3 that is really problematic, and it runs into the same problem as trying to make any sort of self referential universal claim. We're trying to use empirical data to figure out these natural laws. But in order to know that these are all the laws there are, we would need that additional premise. But that's not something we can discover empirically. It's like trying to claim that you've observed every living crow on earth. There's no evidence that can actually support it - only make it more or less probable (or, of course, disproven if you find another crow you haven't observed).
I also don't think we should view these laws as necessary - at least not logically necessary. There are other logically possible worlds with significantly different physical laws than ours. I don't think there would be much of a problem with asserting any kind of physical necessity, though.
2) If yes, does this necesarily prove causal determinism?
Absolutely not. If one of the natural laws includes true randomness (and not just a lack of an observable pattern), then that particular system may be incompatible with determinism. But note I say "may be incompatible". Randomness at, say, the quantum level which gets eliminated at the macro level may still be compatible with causal determinism.
Secondly, we have great evidence that in fact some events are not caused by a predetermined stimuli, but arise in and of themselves.
What are you defining as an event, and what kinds of events are you talking about? It sounds an awful lot like you're suggesting these events are self-caused. But I (and many, many other logicians) find that notion completely self-contradictory.
Basically if causal determinism is true then there are a specific and observable set of stimuli which have led to my sitting here at my computer, and that the exact words I'm writing could be predicted and are the only possible outcome of said stimuli.
An event's being determined doesn't imply that event's being predicted. But predicting an event doesn't imply it was determined to happen, either. All we can do is talk about what we've observed. So even with the best technology possible, it would seem we could only develop one of two propositions:
either
There have been events whose outcome we could not predict
or
Every event so far has been accurately predicted
Note that neither of these propositions can imply determinism or a lack thereof without some serious leaps of faith.