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Answer to Q1: Yes! More precisely, this 2-component vector doesn't know how to transform under translations, so you'd better leave it alone under translations. And it knows how to transform, not under rotations, but under the double cover of the rotation group, called SU(2).
If a spinor doesn't know how to transform under translations, how come we can use it to form a representation of the Poicaré group?
My guess is that in QFT the components of our spinor functions of spacetime, and I think @j_bertolotti didn't make that assumption in his Question
In the homogenous 5d model of Euclidean space given to us by Geometric Algebra, spinors are nicely handled by sandwhiches of bivectors, and translations are linearized as a sandwhich also. So it that context they all play nicely.
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