Yes, definitely bolt it down! Otherwise it will walk all over the place as it starts and runs. And, yes, use some type of shock-absorbing mounting hardware under the compressor's steel feet. Can't tell anything about the size or load capacity of the rubber feet you show, but something like them should do the job.
Also, use a section of high-pressure hose between the discharge of the compressor tank and the fixed shop piping to prevent vibration from passing into the piping. A vibrating tank hard-piped will eventually result in failure of pipe or fittings at or near the tank discharge.