A short disclaimer before answering. Right now I'm doing a Mathematics & Physics undergrad degree and have interest in philosophy. I'm not a specialist in any of those fields. I have plans to tackle the actual math around LLM's a bit further, but I'm not there yet.
To question 1 based on my current understanding (and let's say "beliefs") is that - yes, LLM's are following a probabilistic algorithm to give me the output. The same prompt won't yield the same result, but the same prompt will yield results that are quite adjacent to each other in meaning because of the training. There's randomness in the sense that you can't say which next token will be put with certainty, BUT you can be certain it will be a token that exists in the pool of tokens available. Hopefully that makes sense?
For question 2. Both us and LLMs are constrained by physics. The main difference for me is the fact that at every point of our existence we're constantly changing. Our internal state and bodies are changing based on all the sensory input we get from our environment and the processes that lie underneath. Another commenter in this thread talked about qualia. That's certainly part of it as well (but again we have a definition problem). That's not to say that I think that a hypothetical future machine cannot be devised to handle a lot of these, but the level of sophistication in the biological world is such that I don't think that's realistic. Let's say that even then, that happens, would that really prove that humans work the same way as the machine?
LLM algorithm does not need to be probabilistic, we add probability to make it more interesting, but in principle it should return the same tokens for the same input (which is often desired, and we had temperature=0 for that, even if for practical reasons it was not always working that way). But it is not the point. My point is that LLM inference follows an easy algorithm, but what we get is something different, because weights are part of the algorithm and they are not easily interpretable. So if I ask LLM to write me an essay on a given subject I know what it is technically doing, but writing it as an algorithm different than "convert this text to tokens and than perform billions of simple operations on them to get the next token and repeat" seems hopeless. Because the black box of weights is what matters. In this meaning it is not an algorithm we can find in a book on algorithms. Of course it's a computational process, but the question is, if it is qualitatively different to the one in our heads.