Do you normally adress a gentleman as “gentleman”? I was taught to adress one directly using “sir”: “Good day, sir”, “Hello, sir”, “Thank you, sir”, etc.
Yes, “signore” means “gentleman” (like “señor” does in Spanish) and in Italian (like in Spanish) he can be addressed directly with that title but in English we switch to “sir” when speaking to one directly.
In this sentence the speaker is thanking the gentleman directly: “Grazie, signore”, so it is best translated the way one would thank a gentleman directly in English: “Thank you, sir”.
To call someone just Mister, as in this context would be very unusual in modern British English. Mister needs to be attached to a last name - Mr Jones, Mr Smith. As has been suggested elsewhere here, "Thank you, Mister" sounds like some cheeky lad in an old film. "Thank you, Sir" is polite, as might be used by an assistant in a shop or a waiter in a restaurant.
"Thanks mister" should be accepted. It's less common, but it is used to convey the same meaning as "Thanks sir." I don't know where this weird idea that it's disrespectful comes from, or why that would make it less correct, since there is no hint as to whether the DL speaker is being sarcastic or not. I've only ever seen it used out of genuine gratitude. Some populations among us, like mine, use it more commonly than sir. Furthermore, we don't call our teachers or colleagues "Sir Smith," we call them "Mr. Smith."
"Thank you, mister" is just something that would not be said in British English or, I believe, in American English. Mister, or Mr, is almost always attached to a last name - Mr Jones, Mr Smith, etc.
"Thank you, mister", makes me think of Oliver Twist or some other Victorian urchin. Probably pronounced as "Fank you, Mister".