Learn Russian: Ты выиграл? Как круто!

No, because Что за requires a noun (or adjective + noun) in the nominative case, for example, Что за удача! (What luck!). Since круто is an adverb, we have to use Как (How) to form this exclamation.

Why use Как instead of Какой for this exclamation?

Какой is used when you have an adjective and a noun to agree with, like Какой крутой момент! (What a cool moment!). When you are only using an adverb like круто, you must use Как (How cool!).

Is круто an everyday word that is appropriate for all situations?

It is very colloquial, perfectly matching the English slang 'cool', 'awesome', or 'badass'. It makes sense here since the speaker is talking to a friend (using the informal ты). In a formal setting, you would use a standard word like отлично (excellent) or замечательно (wonderful).

Why is the perfective verb выиграл used here instead of the imperfective выигрывал?

The perfective focuses on the successful result and completion of the action. The speaker is reacting to the concrete, final achievement of the win. The imperfective выигрывал would mean 'were you winning?' or ask if they had ever won in general, which doesn't fit this kind of sudden excited realization.