Moulin Rouge

Acà, a nice Taco spot.

Bouillon Pigalle.

Chez Zena, a nice Lebanese place.

I have never been so don't know how strictly the "dress code" is followed, but the official website addresses your question of attire. https://www.moulinrouge.fr/en/what-should-you-wear-to-go-to-the-moulin-rouge/

"THE PARISIAN CABARET DRESS CODE
A suit, business attire or evening dress: are you in need of some inspiration for what to wear on the evening? We’ll give you the lowdown on what is best to wear to a Moulin Rouge show in Paris!

WHICH ITEMS OF CLOTHING ARE NOT PERMITTED?
Goodbye shorts, Bermuda shorts, flip-flops and sportswear. Leave your trainers at home and slip into your smartest pair of shoes! To fully immerse yourself in the party atmosphere and enjoy the evening like in the golden age of cabarets, it’s important to choose items of clothing that reflect the Moulin Rouge spirit – chic and glamorous partywear. An experience that you will remember for years to come!

WHAT IS THE IDEAL OUTFIT?
Ideally a suit, evening dress or general business attire. Go for an outfit that suits your style but is also that little bit out of the ordinary. Because you’re about to experience a truly unique evening from the moment you set foot in the famous Montmartre cabaret. You can leave your coat or bag (2 euros per item) in the cloakroom, located at the entrance of the Moulin Rouge hall, for the duration of the show. It’s easier to let your hair down when you’re not being weighed down!"

I did not wear a suit. But a nice collared shirt and dress pants and shoes. I was not turned away.

As for "Like an opera"- wife and I go to the SF Opera all the time and I do not wear a suit. Just nice clothes. never been turned away from the SF Opera. Caveat: I do not attend opening nights or gala evenings when Black tie is the norm.

Where to eat: Les Apostres de Pigalle (small plates), Pink Mamma (Italian), Le Wepler (brasserie), and Al Caratello (Italian).
These places listed are all about a 5 minute walk from the Moulin Rouge. Fair warning: Pink Mamma is Insta famous with all the issues that come with that.