Sorry for the late response -- I'm behind on emails.
It looks like you already found the beginnings of a dynamics capability and the tests that we have for that in MOOSE because some of the code you sent for the Newmark method is copied from the existing capability. I agree that we need more tests to demonstrate its usage. We have a post-doc at INL who is primarily working on the dynamics capability. She has already implemented HHT for solid mechanics and tensor mechanics, and we are in the process of merging that in. She will also be adding more wave propagation tests in the near future.
Could you elaborate on what you have in mind regarding dynamic rupture? We have a number of fracture modeling capabilities, including phase field, smeared cracking, domain integrals, and XFEM either implemented or in development. Any one of those could in principle be used on dynamics problems, at least problems that make sense for implicit time integration, although we haven't yet done that.
-Ben