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First let's clear things up: deer have plants they prefer but if hungry they will eat almost anything. Typically, plants with aromatic leaves are unpleasant to deer. Think Russian sage, lavender, etc. Of the honeysuckle vines, coral honeysuckle, Lonicera sempervirens, is said to be deer resistant. Burning bush, Euonymus alatus, is considered invasive in some locales so check with your state extension service. Rutgers has a rating of how deer-resistant common plants are. Again you might want to check your state's extension service for the deer recommendations in your area. Here is Rutger's list: https://njaes.rutgers.edu/deer-resistant-plants/

This can be rather difficult to control. Have you tried a natural predator's urine? Sometimes that can help. Other than collapsing any tunnels that you find, there is probably not a lot that can be done. You have probably tried most methods available. I will include an article for more information on the subject here: https://www.gardeningknowhow.com/plant-problems/pests/animals/vole-control.htm

There are a few things that you can do. This article will discuss your options: https://www.gardeningknowhow.com/plant-problems/pests/animals/natural-homemade-dog-repellent.htm

As long as there is nothing that you are going to consume nearby, then it should not cause too much harm, but there is still a risk of transferring a disease that can be caught by humans. This article will help you to deter cats from the area: https://www.gardeningknowhow.com/plant-problems/pests/animals/cat-repellent.htm

It is probably moles, or voles. These can do extensive damage to gardens. This article will help: https://www.gardeningknowhow.com/plant-problems/pests/animals/mole-control.htm