Is it true that we use "anytime" after one thanks us? Is it its usage?

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Hello Nika,

Yes , in that case it is used as an interjection and means:
"you are welcome"

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It's up to you what you say, you can reply in many ways.

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Yes, you can say "Anytime!" instead of "You're welcome!" or "No problem!"

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When compressed into a single word, 'anytime' is usually an adverb. Like: "He's about to arrive anytime." 

In the sense of "You're welcome!" I would use "any time": 

"Hey, thanks mate!" 
"Sure, any time!" 

There aren't very strict rules for it, to be honest. Except when you say: "I won't have any time until Friday." Then it definitely needs to be two separate words!