If you look in the text, $y$ is not a closed point but a maximal point, meaning a maximally generic point. Then $O_{X_{bar{s}y}}$ is a discrete valuation ring, and so it makes sense to talk about tameness of extensions. Also, surely she's working in the etale topology, otherwise it would be kind of silly to try and represent functors by etale covers, but I didn't actually see where she says that. It's got to be written somewhere, though!
I suppose you're right about a sheaf of sets being a stack with no non-identity maps, but taking this as a definition is exactly as sensible as defining a set to be a category with no non-identity maps.
"torseur sous F" = "torsor under F" = "F-torsor"
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