Open Question: I've got a question about the piece "Greensleeves"...kind of long, especially for people who know music theory?

Okay, I actually have two questions. My sister and I are playing a violin and piano duet out of a Mannheim Steamroller Christmas book for violin and piano. The piece we're playing, Greensleeves, is in G minor, but they don't put in a key signature, well, I guess they put it in C, then they just write in all the B flats and E flats, as well as the accidentals. But the piece is obviously in G minor. They don't do this for any of the other pieces in the book, and they're not meant to be really easy arrangements for beginners. Is there a musical reason for this? Just wondering... haha. Ok, my other question is about the song itself, not the version we're playing. So, Greensleeves goes like this, right? If we're in the key of G minor... G Bflat C D Eflat D C A F G A Bflat G G F G A F D .... etc. Sorry I couldn't write note values, but if you're familiar with the song hopefully you'll know what I'm talking about. Okay, so I'm pretty sure that at some point I've heard at least one recording, or heard it played another way...where they E flat I typed above is replaced by an E natural. Is this another way to play it, or am I just crazy? And if it is, would this be like playing it in the G dorian mode? That would be dorian...right? Or is it just an accidental they put in because they thought it would sound cool... this is assuming that some people actually do play it this way. If you read all this and answer...wow... thanks so much! I just wonder about these sorts of things a lot. I would ask my piano teacher but she's sick and I don't have a lesson for a while.