After illegally gaining entry to the country. That said he might qualify for refugee status, depending on whether the state is willing to recognize Krypton.

I don’t think that follows. You don’t become a citizen merely by being adopted by citizens. You can eventually become naturalized if you fit the criteria, but having entered illegally tends to be a huge obstacle to that.

Um, I don't remember it this way, at least not from the 1978 movie. How would they legally adopt him when his origin was completely unknown (well, the Kents couldn't exactly tell the county officials about finding him in an object from space that landed in their back yard...)? They didn't get into the legalities in the movie, but presumably they just claimed they had him naturally. Of course, this is lying, but as far as the government is concerned, he should be legal.

Was he declared abandoned and legally became a ward of the state for a brief moment?