1. Is the universe actually random at a fundamental level or is the randomness only "apparent"?
On the macro scale, the universe doesn't appear to be random because 'probability' averages out quantum events. The question appears to reduce to: -
"Can a single quantum event be random?" - If so, then the universe is random (and indeterministic).
My own opinion - I still don't know. "Random" is a subset of "unpredictable". Unpredictable events are not necessarily random.
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2. Does QM imply a random universe?
It appears that quantum mechanics actually defines the universe as unpredictable; but it doesn't follow that it's random (see answer 3).
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3. Does "random" just mean "we don't understand this"?
Maybe that is what it means - that we just can't identify a cause
Does the fact that a quantum event appears random really prove that it is? I doubt it, but it may never be possible to prove that it's not.
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4. Does "indeterminism" imply randomness?
Indeterminism, by definition, requires truly random events. Unpredictability and indeterminism are quite different. The former doesn't require random events, the latter does.
