Before designing any support structures, we must know the loading conditions and the constraints.
Luckily, the loading conditions in most automotive scenarios are extremely well known to auto designers. Apart from this, there are also forces due to vehicle motion, such as hitting a bump, pothole, cornering, braking etc.
All these loading scenarios are used in hand calcs and/or FEA models. Using this, as you noted for actively damped mounts, a response curce is generated for the mount and a table is programmed into it's controller.
For passive damping, the loading scenarios are analized, first for safety: Mounts should not have a catastrophic failure in any scenario, and then for NVH/cost reduction. The mounts are usually a complex design, with a dampening material inside them.
Hopefully this can help.