The same as it means in real life, the heir is the next in line to something. In the context of the heir of Slytherin, it means someone from Slytherin's bloodline who had the same affinity for snakes.
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Answer (1): No, Harry Potter is not a prince but he is heir of gryffindor and magical heir of slytherin. Answer(2): (Spoiler alert!) The "prince" in Harry Potter 6 is equivalent to Severus Snape. Answer to answer (2) - in this Severus was half blood prince not prince.
He's the main character, one of the protagonists of the book, and I don't know what you mean about 'role', but I guess it's to defeat Voldemort if that's what you mean.
HARRY POTTER YOU DUMMY!!!!!
"Hedwig's Theme" is kind of the signature song for Harry Potter, if that's what you mean.
That is just a name they came up with for Harry Potter.