Jesse, honored to be included on your list of guys you want to hear from -- not sure how merited it is.
I agree with others that a woodstove heated shelter or fire in front of a lean to is a much quicker way to get warm than planning on a snow cave. Plus I find caves claustrophobic.
The first question is how big of a stove you are trying to run. There is a lot to be said for going with the smallest heated shelter you can get away with and a commensurately sized stove because the size of stove you are running and the size of shelter you are trying to heat determines the tool kit. Plus smaller shelters get more of a gain out of body warmth alone.
When I still had a Kifaru medium stove, it pretty much took a 24" bow saw and minimum 20" handled axe to keep it running in the winter. It's not a very efficient stove though. Larger stoves in 8 man or larger shelters, same thing. Right now the only back packable stove I have is my homemade "little pig" which gets the GoLite Utopia very warm and keeps the Seekoutside BCS warm enough. It can be run pretty well with the 6" Gerber slide out saw and little 4"x1/8" blade D2 utility knife I carry. There's an in-between setup that has me carrying an 8"x3/16" bowie OPC gave me and a japanese tree trimming saw. I have also carried the cold steel tomahawk a fair amount as the absolute lightest implement available that allows for a two handed overhand chop.
I must say that I'm in the camp of being biased towards batoning for safety. In my days as a forest fire fighter, I saw first hand what can happen when you mix exhaustion with pulaskis and chainsaws. It takes a lot of very purposeful concentration to stay safe with tools when you are exhausted.
Open fires in front of lean tos are going to be less efficient at heating so you'll want to bias towards the larger tool kits.
Once the snow gets deep enough, the Voile Telepro is always along. It's the real deal for moving a lot of snow quickly and an improved tree well that offers a heck of a lot of shelter can be had in about 5 minutes.
Oh... don't forget about insulation underneath you. Be it closed cell foam in the form of a camp chair or foam pad, or tree limbs or whatever, you need something between you and the snow.
For fire starter, 9 times out of 10 a 2x2" square of bicycle innertube is all I need to get a fire started. Trioxane handles that tenth time. I must admit to using a *lot* of trioxane from time to time in Big W's neck of the woods. A pocket rocket was used as a blowtorch once too. I'm not going to say that a bottle of outright fuel is a bad idea in those conditions. It's not quite that wet where you are though Jesse.
