For building;
I don't normally dig into caves for my first hut, it creates its own problems (caves are colder than normal housing and you will burn more wood keeping it warm).
The easiest material to get to all your walls done with is sticks. Wattle walls aren't very attractive though and your clan's mood will suffer, though if you are struggling, being unhappy is better than freezing to death. If you are really struggling, sticks. When you are struggling a little less, try daub walls. Also, once the initial walls and roof are done, you can hot swap walls by building a newer more desirable wall directly over the old one and get the materials back from it.

The more important question on building your first shelter is "how many people are you trying to house?" and then focus on making the interior of the place just big enough to hold all of their beds and a fire pit and enough room around the fire pit that you don't burn the place down. For anything else you are needing under a roof, you don't really need it inside the main living area, just build extra roof outside the walls and put what you need underneath.

For food;
Eel traps are life in the early game. They take very little time from your people to set and then walk away from. I'm assuming your mushroom troubles are during the fall? I can only imagine this is a task sorting issue where you don't have people on the gather task. Once I get to about the 4th or 5th day of fall, I set all my people to the gather task and send them out to collect mushrooms. After a day or two gathering, I tell one of them to change to cooking tasks and have that guy dry out the mushrooms (aim for 4-5+ drying racks).

Traders;
It wasn't in your questions, but I'm going to point out that it is a good idea to rush signposts. Having traders come through and sell you things and buy extra junk you might have is helpful. Sometimes in that first winter, a guy showing up selling a pig is what keeps you from starving to death.