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.
They are different, "nice to see you" implies that you have already met or seen the person before. "Nice to meet you" means that it is your first time being introduced to that person.
"Nice to see you, it's been a while since we last met."
"Nice to meet you, what's your name?"
Another phrase that is commonly used is "Nice to see you again". It is interchangeable with "Nice to see you".
"Nice to see you again, it's been a while since we last met."
Best Answer
"Where are you from" is more common (at least in U.S. English). In my opinion, this is the preferred option when meeting someone new.
The difference is a bit more emphasis on the verb "to be/are" instead of the action of "coming/being from" somewhere as is common in other languages. The meaning is identical though.
1) "Where are you from?" implies that you want to know what city/state/country they consider "home," and that you assume it's someplace other than where you are right now. This may be confusing, since where someone is "from" isn't necessarily where they live. It also can be embarrassing to assume, for example, that a person of a certain ethnic background must "come from" some other place. It might be a good idea to separate the idea of "what is your family/ethnic background?" from "where do you live right now?"
2) "Where do you come from" sounds kind of awkward and outdated. I think this is because it includes the present-tense verb "do" in a question about a past-tense action (coming from somewhere). This is perfectly understandable and you can say this if you prefer - I'd still suggest the other way though.
3) "Where did you come from" would be correct if you want to know "where were you immediately before you came to this location we're at right now?" This can also be used to express surprise at the person's sudden arrival or unexpected actions.