Chickens and moles are not infested by the same kind of mites. You should be giving your dogs something for flea and tick control, anyway, because there are many sources of these parasites.
The only sure fire way to get rid of moles is to poison them, but if your yard is desirable habitat, others are likely to eventually move in after you kill the one(s) you have now. Keeping chicken feed and bird seed out of their reach will not help keep moles away, because moles are insectivores. They don't eat seed or grain. I would suggest you try to tolerate the mole. Maybe get a field guide, learn more about them, perhaps identify the species you have. They really are interesting creatures.
I really don't think plugging up the hole or stuffing anything down the tunnel will have any impact. Moles are built for digging and will simply dig themselves out in some other direction.