Learn English – what is the difference between ‘come off’ and ‘take place’

American native English speaker here. My gut check says that for the first pair, there is a extremely mild, extremely subtle difference in each case. They are sufficiently interchangeable, that even I, who am pretty big on subtle distinctions, think you can get away with exchanging them freely.

I am from China.

I come from China.

If I use the first formulation, I am possibly stating a perfectly neutral fact. It may be in response to a question about where I'm from.

But there's also a possibility that I am making a causal statement about why I am the way I am or why I see the world the way I do.

If I use second formulation, it is guaranteed to be the non-neutral connotation. It's never just a statement of fact, it's a statement of identity and culture and it's in a context where it's explanatory and causal. "I come from X, and the way we do/did things there is...." It means something like "who I am arises out of my belonging to the culture of X."

Additionally, the word "come" implies "somewhere not here" -- after all, to have come here you must have come from somewhere -- in a way that the version without doesn't. So in the question forms,

Where are you from?

Where do you come from?

the first is neutral in assumption as to whether or not the person asked is from here or elsewhere. The second implies, "Since you're not from here, where are you from?"

Because of that implication, "Where do you come from?" can be either more or less polite than "Where are you from?", depending on context -- particularly on whether or not the person asked has identified themselves as being from somewhere else. Asking someone who has identified themselves as from elsewhere where that elsewhere is is considered to be showing a polite and even flattering interest in them. Asking someone who has not so identified where they came from -- when they might have been born right here -- can be tantamount to calling them a foreigner, and suggesting they are unwelcome.

There is no difference in their meaning in the examples provided: wonderful opportunities are rare ... wonderful opportunities present themselves rarely...one does not happen upon wonderful opportunities often.

"Get" is by far the more common.

But the two locutions could have very different connotations in other circumstances.

How did you come by that fancy camera? versus Where did you get that fancy camera?

"How did you come by" could be understood to be an insinuation that the person with the camera had obtained it illicitly, whereas the question "get" is neutral; the questioner who uses "get" might well expect an answer such as "My uncle gave it to me as a graduation gift".