The general answer to all your questions is no. You can't build a house on any plot of land you feel like. I'm sure someone with more knowledge will correct me, but the basic route to building a house on a piece of land, as opposed to buying an already existing house, is:
1) Buy the land. This generally involves you and a broker but it could also be done through private parties (i.e. from you to me). In either case there is a record of who owns what, the amount they paid and, most importantly, a record with the local government of who now owns the land
This last step is crucial as it prevents an agent from selling the same piece of property to different people or someone building on someone else's land.
2) Find a home builder. Once you own the land you have to find someone to build your house unless you're going to do it yourself. Regardless, this involves filings with your local government to make sure the building meets the local requirements for water and sewer (whether on the land or through the public service), certain structural designs and so on. Since every municipality is different, some are more lenient than others but you still have to notify them you're going to build your house so they can determine how much tax you will pay on it (again, depending on the municipality. Some places don't charge tax on property, others do).
The 4th Amendment does not enter into this in any way. The 4th Amendment only comes into play once you have your property. The police can't walk in just to see if you're doing anything wrong.
To sum up, if you're building a house anywhere in the U.S. you have to file enough paperwork that everyone will know about it. If by chance you were able to build a house without anyone realizing it and were then found out, you'd have a lot of legal issues to take care of.