In regards to getting rid of spider mites:
Try an oil spray first - they are like microscopic aphids and they suck sap from the tree. If you only have a couple small trees then you should be able to saturate them with an oil spray. The good thing about oil is that it suffocates the insects - so they can't really develop resistance to it. If you have more trees or the oil spray doesn't get rid of them then switch to malathion, orthonex or bayer shrub and tree care, or look for a specific miticide. Those are systemic chemicals which the tree will metabolize and then when the mites such sap it will kill them. Spider mites have about a 10-day life cycle so you need to re-spray the tree two more times at ten day intervals to catch mites that are hatching after you sprayed. Alternating with two or even three insecticides helps ensure that the mites don't build resistance. Make sure you spray everything that you have, not just the infected tree. They are wind blown and can easily re-infect a tree. If you have large conifers like Italian Cypress, pine or similar in your neighborhood that are infected you will have an ongoing problem.
Overhead watering with a good spray of water discourages small infestations from getting large by washing them off the plant, but once they get established the water method doesn't work. I would recommend that you overhead water your tree at least a couple times a week, not misting, but a good overhead spray to rinse things off.
regarding Shimpaku and scale - I've seen small white scale infect shimpaku. They're not particularly damaging to the trees in my experience, they usually attach the weaker growth and shaded areas. They look like little spots of sap. If you flick the branch you might see small white flakes fall off. The same type infects procumbens and it seems to prefer procumbens but it generally only weakens branches a little, it doesn't kill them like spider mites do (at least in my experience.)