In my current WIP, I have a similar situation. (As in a multilingual situation where people speak several different languages.)
When two characters that are knowing the same language are speaking I do it in the main narrative language even if they are speaking another language. So, for instance, in your case, if two people were speaking French, I'd still do it in English.
I also don't comment on it with a "said X in French". I just do "said he/she" as if they were speaking English. My assumption is that if they both know the language as natives, they'd be as native in the main language of the novel as in their specific language. The experience of the reader is central here.
You may still use language/culture to influence the content and focus of the dialog to make it sound more "French", as long as you don't do "Inspector Clouseau". (Maybe for comedy, but probably not even there...)
The same goes for thoughts.
When someone that doesn't know the language hears the dialog, I'm mostly doing it in the "other language".
I've also done the first of your examples. (The "X said in French" variant, but that's when the POV character knows the used language... it is as "English" to them, though they are actively choosing another language.)
The big difference is that in my case the other language is a con-lang, so I have to do stuff that will clue the reader in on what's going on anyway. There can be no assumption of reader understanding at all... (And to be honest, my characters find it rather rude to do a lot of dialog people around them cannot understand. For obvious reasons, in most cases, everybody uses a language everyone knows...)
Your "greeted formally" could be one way, or the POV person thinking something about the person or they shaking hands and bowing or other actions that would show what's going on. Another thing would be to use foreign words in a way where action or reaction gives away their meaning. "Let's go to the xxx to buy some groceries." "I need a drink, let's go to the xxx." Or even better, just go there and show it.
If you've chosen English as the language for the novel, the expectation is that other languages will come in easily inferred phrases or insignificant ones. For instance curses, greetings, and other non-essential text.
Of course, you're always free to choose both English and French but from a marketing/translation perspective that one will be a much harder sell to any agent or publisher.
It's also ok if not all your readers will get all your language quirks. As long as it doesn't get irritating or influence the understanding of the plot too much.
Finally, don't italicize "foreign" words. This is a great video on why it's a bad idea. Instead, use italics when you want to stress some words or text. Also, look into how to do dialog. I'm not sure your format is a common one. It's better to figure that one out now than when you have 100k words to go through and fix.