Is each hat awarded multiple time or is it just awarded once?
No. And why do we need to earn it multiple times? We can't wear multiple hats at the same time. Then why do we need to have same hat multiple times?
Today, I answered questions with 5+ score more than one time but I have earned Where is He?! only once.
Nope. Why do we need same hat with multiple times? It doesn't write in FAQ but I think this isn't an option. But you can earn the same hat for different stackexchange sites.
Nope -- you'll just have that hat one time. However, this also means you don't need one question at +5, one at +10, and one at +15 to earn those different hats -- it's cumulative, so if one question gets to +15, you will earn all three hats.
That said, if you're active on multiple sites, then and only then can you earn the same hat repeatedly. Given that your hats on SO don't follow you to MSO, this is a great way to extend your hat collection and try out other sites!