I would say they will understand, may charge a fee similar to bringing your own wine but I doubt it.

They serve only A La Carte and a fixed menu. They open from 11 am.

No in oil

Nope

Outside yes

There is no dress code, but the atmosphere is definitely business casual.

Office people during the day, and post office hang out at night

No there is not

Yes, and it's very recommended, weekends and holidays are super busy

We did pay for parking.