Is greek style yogurt necessary? I find most of the greek yogurts have emulsifiers in them such as gelatin or carageenan and I don't care to eat...

Gelatin is actually a common ingredient in yogurt in general, not just Greek yogurt. It usually seems to be the cheaper brands that use it. In addition to the ones amy mentioned, Liberté is another brand that doesn't contain emulsifiers. But as others noted, you can also just strain regular yogurt through cheesecloth to thicken it up.

Not sure what brands you buy but I just checked and Fage, Chobani, Stonyfield and Wallaby and they only contain milk and cultures (at least according to their websites). I wouldn't want emulsifiers either.

Greek-style yogurt is just strained yogurt. You can do this with regular yogurt by letting the yogurt drain through some cheesecloth until its is nice and thick, ~4-6 hours sometimes longer. So yes, you can use regular yogurt just drain off the whey or the texture of the frozen yogurt will be too hard and icy.