Although the British stopped using the past participle gotten about three hundred years ago, the American colonists and their descendants--especially in New England--still tend to use it.

Some English teachers have tried to ban its usage to make American English conform to British English, especially during the nineteenth and early twentieth century when there was a movement to purify English. Others are just not used to its use because it is not used in their region and hear it as an error.

Ultimately, language is convention. If you are writing for a formal audience outside of New England, you might want to use the simple past form got instead. It is like the dictum to never end a sentence with a preposition because that is something some people just will not put--ummm--up with which some people just will not put!

Yes.

For example: "Since I last saw you, you have gotten big!"

Gotten is correct, and very old. In England many people wrongly assume that gotten is a modern Americanism, but the truth is the English more-or-less stopped using it, and have forgotten (!) that they used to use it.

That said, "gotten" isn't good English. In most cases other, more precise and meaningful words should be used in its place.

While "have got" sounds wrong to American ears, "have gotten" can usually be replaced by "have become", and "have been able to" or "have had the chance/opportunity to" would make better sense in other situations.

"You would have got along with him" is proper English.