If you are not a US citizen, not a recognised tribe such as Cherokee or Navajo, and have no citizenship, it is a crime for you to over stay your time limitations on your entrance visa. There may be some discerning rule concerning students, but I argue against it unless you are paying the full amount for the education and are not taking some resource away from an american citizen. If your status here requires a green card (as long as i am unemployed, we have no need for your family here in my town) and you have been here over 1 year (unless laws have changed) you would have to return to wherever, and re-apply as long as there is not some bogus law stating you can complete school first.
Yes, so if you have a visa, you are allowed to stay as long as your visa permits. Also, some visas are extendable.