How should I set up a two-layer dome tent for 2/4 people in deep snow (waist height)? Do I need some additional equipment (snow shovel)? Insulation under the floor (further than the sleeping pads)?
Last weekend after a snow storm in Quebec I camped in the conditions you describe. Around 1 meter of snow, -12C/10F during the day -24C/-11C during the night.
Make a layer of spruce branches ~15cm or more. Also, use a closed-cell foam pad plus an inflatable pad. I slept in US Army bivy sack, US Army cold-weather sleeping bag. This set-up will keep you reasonably warm. To keep you comfortably warm, sleep dressed in many layers, wear balaclava and a lot of socks, otherwise your feet will get really cold. Exercise before going to bed. You might want to put some heat packs in your socks.
Use a snow shovel to dig a hole for your fire about 2.5 meters/8 feet in diameter, plus a place to sit and sleep. Otherwise your fire will melt a hole and all the warmth will radiate up.
Bring a pee bottle. It's not fun to go in the dark cold forest and then try to get back to a cold bed.
Do not underestimate the amount of wood you will need and time it takes to gather it. Chop a couple of big dead trees and split a few logs to make a nice fire. Bring an axe and a foldable saw. Put a bright lanyard on every piece of equipment.
Also use large snow shoes, you will have to carry a lot of weight, if you don't, you will not get far.
Don't go alone. Bring some medical supplies to treat small wounds. If you bring alcohol, don't bother with beer, bring something stronger. Otherwise bring some hot chocolate, it is easer to clean then tea.
Winter camping is easy and fun once you find the way to do it.
I usually stomp the area with snowshoes until it's reasonably packed, and that's good enough. If it's windy and very cold I may dig down 3-4 feet for protection from the wind. (If you're in the mountains you should have a snow shovel for safety.)
A thicker sleeping pad (or two pads) will definitely keep you warmer. When you pack up in the morning, you can usually see an indention in the snow where you slept -- your valuable body heat melts the snow under your sleeping bag.
A snow shovel will make your life a lot easier, yes - it gives you the option to dig out snow to use as a windbreak, and to lower the tent a little into the snow, reducing the wind.
Digging down a little also lets you reach more packed snow, which makes it easier to provide a flat surface.
You shouldn't need any extra insulation under the floor - although if temperatures are reasonably warm (not far under freezing) you may want to use a good groundsheet as you may melt some of the snow directly underneath you and you don't want that soaking in to the tent.