Welcome to permies Josh.
Not knowing where you are, or the climate, etc. it's hard to give a good
answer.
If your
water table is high, or the soil drains poorly,
underground might not work well.
You don't want them in wet soil.
I once saw what looked like an igloo, but made with adobe rather than ice blocks. The owner stated that it kept a constant temperature all year long. I believe him, as I grew up in California's Spanish Mission territory. Whether it was 30 degrees, or in the 90's outside, inside the mission buildings, it always hovered around 68 degrees. They understood building for that climate, but unfortunately knew little about CA earthquakes!
As far as stopping them from digging, I think that depends on the individual dog.
Tell them if they dig deep
enough, they'll end up in China.
Then tell them there are 100's of recipes for "woking a dog". LOL