if you find an abandoned homestead can you take it over & build?

No, In the USA, Squatters can't move onto or into property and claim ownership of it. If they could, people would move into the woods on a large ranch or farm and just stay out of sight and then claim ownership of the land.

There is a situation called Adverse Possession, but ownership does not change hands, and normally it applies to things like a fence line that has been standing for years over the property line and no one did anything to correct it.

The owner must be aware you are living in his house or on his property and he does nothing about it, then you can make a claim for Adverse Possession. That does not mean you own it, it means you can continue to live on it. Once you move, the possession reverts to the owner and a new fence can be put up on the correct property line. A new person living in the house or on the property would not have been there enough years to make a claim for adverse possession.

No. Someone owns it, even it is a local government entity like a tax assessors office.

Edit: Then I assume that no one has been payig the taxes on it, and the possession will revert back to the tax assessor, who will eventually sell it at auction. With these old abandonments, the tax assessor treads slowly, trying to see if there are heirs that are going to claim possession.

It is extremely unwise to take up residence here and try to make it livable again or build, that is money you are going to throw away. Eventually, some person or some entity is going to step in and evict you.

There is a statue of limitations before you can file a deed. If a owner of the property came and so you made improvements they could order you off the property, normally it takes 10 years to make such a valid claim.

No, someone owns the land. You need to find the owner and buy the land.