It's very hard to grow lawn in shade. Can you limb up trees?

If they're Norway maples, forget it. Grass won't grow.

Sod is grown in full sun conditions. Probably bluegrass. You want fine fescue and perennial rye as more shade-tolerant. Probably have to seed to get these grasses.

Squirrels are noted diggers.

If you can't fix the shade, consider groundcovers, patio, gravel.

I've lived on Jersey clay most of my life and it grows lawns just fine with enough light.

I agree that sod won't grow without sun. I am not sure about the arbor vitae. It sounds like they are not doing well due to the soggy ground. Some plants can "drown" if the soil doesn't drain well. There are several things you can do:
1. Improve the sun exposure by limbing up trees. Please do not cut down mature trees unless there is a safety issue. Shade is a wonderful thing, easy to get rid of, but difficult to add.
2. Improve your soil. Coffee grinds, compost, shredded leaves, lasanga created beds, ... can all help with breaking up the soil. None are immediate cures. The soil forum can help with more if you are interested.
3. Plant shrubs that like shade and soggy ground.
4. Investigate where all that water is coming from. Perhaps there is a drainage not working properly, or a buried stream or your neighbors sump pump or ....

Also, we have had an unusual amount of rain this fall, and the ground is more saturated than usual.

Hope this helps somewhat.

I'd recommend getting your lawn core-aerated next fall and then topdressing lightly with peat moss and lime.

Well, that stinks. Seems like you lost a little chunk of change on your lawn and arborvitae. Where did you purchase it? If it is less than a year old, you might have a warranty on it. Did you save the tags? Did you use a landscaper? He should have known better.

All good advice so far. Why is your ground wet? Are you near a wetland area? a stream? the bottom of some runoff or a high water table? You need to troubleshoot that or work with it.
For evergreens that like shade, look at yews, hemlock (you might have to spray that for wooly adelgid) and broadleaf evergreens like rhododendren, hollies. But before you invest more, investigate your drainage. Will the roots be sitting in standing water? Can you fix that? Because if the answer is yes and no, you will have to choose accordingly. Also, do you need the privacy in winter or do you mainly just use the yard during the more mild seasons? That could open your choices a little. Don't forget to consider if deer feast in your neighborhood.
Although it sounds like a water / light problem, don't forget to check your lawn for grubs. They like to eat root systems and a sign of them is grass pulling up easily.

You will get visual privacy from shrubs but you will never get sound privacy. For that you need something solid without holes and resistant to sound transmission. Quick illustration: you are in your house with windows closed - open one window and you can hear everything even though most of your surroundings are more solid than a bank of trees and shrubs.
So if you want to try to mask sound, try berming the edges of your property and building a solid wall. If the wall is a couple of feet high - higher than an exhaust pipe, it will at least bounce back some car noise.
Then there is white noise. You can use that to mask other sounds. A good example of this for the garden is the sound of water flowing or splashing.
OK? At least that is what I remember from my acoustical engineering class.
The holes could very well be squirrels hiding or looking for their stash.