What exactly is the Advantage Law?

This is a popular misconception, there is actually no Advantage Law, but is covered within Law 5 “The Referee”

Which states that the Referees shall allow play to continue when the team against which an offence has been committed will benefit from such an advantage and penalises the original offence if the anticipated advantage does not ensue at that time

Details can be found on here.

When has the ball crossed the line for a goal?

The method of scoring is covered by Law 10.

A goal is scored when the whole of the ball passes over the whole of the goal line, between the goalposts and under the crossbar, provided that no infringement of the Laws of the Game has been committed previously by the team scoring the goal.  

Details can be found on here.

Why is handball sometimes given and not at other times?

There is a lot of misunderstanding about handball, which is covered within Law 12 “Fouls and misconduct” which lists

“handling the ball deliberately (except for the goalkeeper within his own penalty area)” as a offence which is penalized by the award of a direct free kick. 

Details can be found on here.