Law is a subset of social norms. Norms, by definition, are norma**l. It is normal to drive on the right side of the road in America. If you were to drive on the left side of the road, you would be breaking from the norm. Likewise, murder is prohibited (okay, the state can murder, but for the rest of us, we are not allowed to do it). You cannot have two norms simultaneously stating that murder is allowed and that murder is forbidden. Only one can be a norm.
So, fear not. In libertopia, there will not be conflicting laws in the same geographic location.
Oral contracts are just as valid as written contracts. The problem with oral contracts is that they are harder to prove what the agreement was, as there is no written record.
Here is what I wrote in another thread:
The NAP is pretty easy to remember, but it’s been written down in lots of places. Regarding courts, I suggest that you read What Law Is and A Praxeological Account of Law by forum member Clayton.
I highly recommend The Ethics of Liberty by Murray Rothbard. I do not agree with him regarding natural rights, but the vast majority of what he writes is spot on, as he almost always uses the NAP as his starting point. There is a chapter Children and Rights that should probably answer your questions, but I suggest you read the whole book. It’s not that long.