I used a mesh size that is about the thickness of the body to be meshed. The rubber is 3 mm thick and I chose a mesh size of 2 to 4 mm. This results in a pretty coarse mesh which runs quickly. However, if you are concerned about the accuracy of your results you should really do a mesh refinement study. This is were the mesh is made progressively finer until the results do no not change from one iteration to the next.
For structural simulations I always use a second order mesh. A first order mesh is quicker but will often result is stresses that are significantly lower than they should be. Please see this forum post for more details.
https://www.simscale.com/forum/t/when-should-i-use-a-second-order-mesh/25899
Material damping can be configured in the material properties section. You will need to source the values yourself (with a Google search).