Eh, growing up near Seattle, we bought satsumas every winter. Where I'm at now, I can only find mandarins, which have a bit tighter skin vs the super loose skin of satsumas.
I actually didn't know what a mandarin or tangerine were until moving away and not having access to satsumas, so it really depends on where you're from in this case. Now that I live away from easy access to satsumas, I understand how confused others must be since they're nowhere to be found in local stores.
It is the 'object counter' used when you want to count items. Basically it means things, but only in the context of counting. Some more info: http://keytokorean.com/classes/beginner/how-many-counting-stuff-in-korean/
In Korea, there are no American oranges, so they just call them oranges. You come back to the US and there are dozens of regional names for them (I even heard somebody call them "cuties" and I had no idea what she meant!) I have never heard of satsumas before, but I don't think tangerine describes "gyul" (no Korean keyboard) quite right.
I think of tangerines as lighter in color and with more seeds (and more difficult to peel). I never heard the word satsuma before this discussion thread.
I definitely agree that they are different from "American oranges" with a thick peel and a lot of white stuff (very technical term) all over the little sections. 귤 are very easy to peel, have very few little strings on the sections, and are much smaller than the gigantic "American oranges." They are also mostly seedless.