The third part of the question is easier to answer than the first two: whatever distribution you get from the estimates, you can use it as a prior for further Bayesian updates.
Strictly speaking, you should distinguish between a probability of some event, and a distribution over models. Bayesian updates don't really work on probabilities directly, they work on model distributions. For any real-world prediction like this there are lots of relevant models, and a single probability doesn't specify enough information to do Bayesian updates.
For the first two parts, it's a lot harder.
If the estimates were part of a prediction market that you have reason to expect is "efficient enough", then you should expect that you can't do any better than using the average of the probabilities. Otherwise you could make a spread of bets that has positive expected value.
Metaculus is not such a market, and neither the median nor average have this property. There may still be some more complicated function of the predictions in the pool that does have useful properties, but I suspect finding one would be a major research project.
In the absence of such a justification, you will have to settle for the boring but still useful answer "it depends upon what you're doing with it". If you would benefit from having roughly equal numbers of people who make these predictions more and less optimistic than yourself (no matter how much more and less), then the median makes sense. If the degree to which they differ from you does matter, then some sort of average makes more sense. If some other property is desirable to you, then use some other statistic that reflects that.