Learn Italian: Non hai mai visitato questo museo? - Have you never visited this museum?

In Italian, you need a "double negative" to say "never". You must put non (not) before the verb, and mai (never) after it. If you left out the non and just asked hai mai visitato?, it would mean "have you ever visited?".

Why is mai placed right in the middle of hai visitato?

When a verb has two parts, like the past tense here (hai + visitato), short adverbs of time like mai (never), già (already), and ancora (yet) are usually placed right between the helper verb and the past participle.

Could this sentence also translate to "Haven't you ever visited this museum?"

Yes! In English, we often ask "Haven't you ever...?" to sound a bit more natural. Italian uses the exact same structure (non ... mai) for both "Have you never...?" and "Haven't you ever...?".

Why is it questo museo instead of questa museo?

Museo is a masculine singular noun (it ends in -o), so the word for "this" must also be masculine singular, which is questo. If you were talking about a feminine noun, like città (city), you would use questa.