I like fixed as the front end is lighter and it does not move around. Carbon if you are willing to spend the money.
Single speed means no derailleur to break. And smaller chain ring and bash guard gives you more clearance.
I don't think you want a real wide tire. I would think like a 2.1 - preferably tubeless. If you start riding 2" pipe you may even drop down to like a 40mm and for that you buy a CX tire.
I don't practice skinnies but I do know that a low pressure is better for crawling over wet roots and rocks.
As for winter bike it does not snow where I live now but I think this would make a good snow bike. And if they salt you don't have expensive components getting eaten up. The bike below only takes a 2.1 in the rear but you can find them up to a 2.4.
As for buy cheap new my advice is no. Cheap new you are going to get wheels that flex and bend on the first crash. You get racers that upgrade every few years and have worn out the front fork - replace it with a rigid. You need a bike that is going to be solid under you. Nice wheels alone are $400. I would budget $800.
This is the bike I bought for learning technique. More to learn to bunny hop than ride skinniness. This was $850 used but that fork is worth $300. Not a cheap bike but it is a bike that would be hard to break. And I ended up riding that bike for a lot more than learning technique - it is fun to ride.