What’s angio-?

Is it okay to say "gonna" instead of "going to" even in the formal speech?

In formal speech, it is okay! Within natural speech and dialect, it is recognised as "going to." In formal writing, it would not be acceptable. Ex: By 2024, we're gonna launch the latest model of our product. => formal speech = By 2024, we're going to launch the latest model of our product. => formal writing

Will the meaning change if it was "must" instead of "should" here?

It would change ever so slightly! "Must" would work really well here, actually. But there is a slight difference in nuance in this context! In this sentence, "must" would mean that you're certain or expecting the other person to be happy. "You <should> be happy with this" is, like, you've done enough work to be satisfied, so be pleased about it, and you may not think the other person is satisfied yet. Ex: You should be happy with the work you've done. You've worked hard. = You must be happy with the work you've done. You've worked hard.

What’s angio-?

"Angio" is short for "angiogram", which is a medical test that takes pictures of your blood vessels to help make a diagnosis.

What does "op-ed" mean? Does it refer to an "opening and ending"?

"Op-ed" is short for "opposite the editorial page," and is colloquially sometimes known as the "opinions and editorials page." It's an opinion piece in magazines that is usually written by someone unrelated to the editors of the magazine. Ex: A writer I admire wrote an op-ed for the New York Times. Ex: I read an interesting op-ed recently that talks about how in the future, education might become entirely digital.

What does "fix one's charges" mean?

To answer this question, it's important to give context to the scene. Cobb (DiCaprio) has been charged with the murder of his wife. He did not commit this crime because his wife committed suicide and wrote a note saying that he did it. When he says "fix one's charges" he means that these people have special connections that could remove the murder charges against him. It isn't really common to say "fix my charges." A more common way to say it would be "remove my charges."