Yes, possibly. Publication bias is a known problem when using prior work (see [1] for some references on the problem and potential solutions).

One of the strengths of Bayesian methods is that you can use a less informative prior to model your skepticism of prior work.

[1]: https://www.sciencedirect.com/topics/medicine-and-dentistry/....

Yes, but no more than a non-Bayesian’s. The difference is mainly that the Bayesian approach makes more of its assumptions explicit.

Yes, it should affect anyone's conclusions unless they know exactly how much publication bias there is and then account for it

Depends how you model it. I would say not taking into account the publication bias if you have ways to measure or at least estimate its distribution is a bigger mistake to make.

That's why it's for constructing a prior that you change later; your study itself should update the prior in the direction of being more correct. But it's not like you can do better than "the best available prior" (by definition).