How do we get rid of the voles, repair the lawn and cover the tree trunk that is exposed?
I was so sorry to see your pictures. It does look like vole damage. The lawn will be easy to repair but your beautiful tree is another matter. Voles can be trapped using similar traps that you would use to catch mice. They love peanut butter and that is what seems to work the best for traps. The box traps with sticky interiors will work well also. You could also try some of the poisons that are good for mice or rat control. Just make sure pets do not get into any of those. This winter, with the early snow cover and so much of it, provided a great environment for voles and mice. I am sure many will find damage like yours.
As far as the lawn repair that is fairly easy. If you just rake out the tunnels and trails, the roots of surrounding grass will move in quickly and fill the gap. If needed you can overseed any areas with a good grass seed mix this spring. Your tree may be a bigger problem. There are no products to cover the missing bark areas. Your tree may be damaged beyond repair. The layer just under the bark of the tree is where the tree transfers all the water and nutrients up to the leaves. If most of that is gone around the base of the tree your tree may not survive. You can wait to see if it produces leaves and if it will get enough water movement up the trunk tissue.
Thank you for your question. We are always glad to help where we can with all of your gardening needs.