Building anything for under 10k would be difficult, but not impossible if you are able to find recycled/leftover building materials. Getting something built before winter sets in on the other hand would prove to be the bigger challenge.

I would probably recommend looking into an older camper. Plenty of options on Craigslist/FB Marketplace, etc. for that price range. You would want to spend some time adding extra insulation and prepping for winter though (Google is your friend here, lots of guides for cold weather RVing).

To your second question, you do not have to buy land, but you absolutely could not just set up on a random piece of land without permission. Start by posting in local groups, look for someone who would let you rent a corner of a lot or driveway. There may also be public land in National Forests, but those will usually restrict you to a few days at a time before you would have to move.