> 3) LLMs do not have a mechanism for sampling from given probability distributions. E.g. if you ask LLM to sample a random number from 1 to 10, it will likely give you 3 or 7, as those are overrepresented in the training data.

You can have them output a probability distribution and then have normal code pick the action. There's other ways to do this, you don't need to make the LLM pick a random number.