Does the father have equal rights to see his children before the courts stipulate visitation? Is a verbal agreement viable between the two parents? Can the father deny the mother to take the child if she is trying to take the child early?

1) technically yes, although it is hard to enforce. Typically the parent with physical custody makes the rules.
2) in short, no.
3) sure... if both parents are legal parents and one has the child, the other cannot physically force an exchange. But it's a bad precident to set in an arrangement that is likely to last 20 years or more.

How early is early, hope you are not talking 20 min or something

Until there is a order in court, there is nothing to "enforce" so the parents are sort of at the whims of the others.