1. I believe it is possible, using "inertia relief", but i highly recommend against it. The problem is that the software compensates for any static movements, but it does not show you it is doing this. If there is something wrong in your model it will likely be compensated for and you will never notice. It is much better to think about good constraints, even though it may need some more thoughts. In your case, maybe you can use a symmetry constraint, or otherwise use the 3-point constraint system where 1 point is constrainted in all directions, 1 point in 2 directions, and 1 point in 1 direction, so that the part is free to deform in all directions while still being fully constrained.
2. If the von Mises stress is higher than the yield stress, then yes you will have plastic deformation... but this may not be the best way to check. You can also output "plastic strain", and if that is larger than 0 you always have plastic deformation (as the elastic strain is excluded from this variable). Another way to do it of course is to unload the model and check deformations, but that will add calculation time to your analysis.
Here is a good PDF on the subject, posted by Giulio Fraulini: