In my opinion, using DST usually adds unnecessary complexity to problems that can be sufficiently solved in a Bayesian framework. Then again, I think that the same thing can often be said of descending from a Bayesian to a Frequentist approach, which is to say that most problems are simple, and properly using any framework is enough to get a good answer. See neq1's post that inspired my original comment.

That said, I work on problems that I have solved both from a Bayesian perspective and from the perspective of DST, and I have found the former lacking. There are at least a few problems that I feel like DST is much better at. If you search Google Scholar for Dempster-Shafer and look at results in the past few years, you'll notice a really clear trend for using it to extract information from noisy sensor data. That's what I use it for, and seems to be a strength of DST.

As to your second question, I think it is in the realm of possibility that Bayes can be used to construct DST, but I don't know how and if it is possible, it is certainly more difficult than going the other direction. In some sense, DST is meta-Bayesian, because PDFs of PDFs of priors can be specified, but doing that with a strictly Bayesian framework misses the set-theoretic nature of Dempster's Rule of Combination, and results in a weaker theory, that among other things, still doesn't handle contradictions any better than Bayes does.