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    $\begingroup$ The paper "Set Theory of the Plane" by Miller (see math.wisc.edu/~miller/old/m873-05/setplane.pdf) contains many results concerning the covering of the plane. $\endgroup$
    – Mohammad Golshani
    Commented Oct 22, 2013 at 6:53
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    $\begingroup$ For your question 1, the answer is no. If $\{X_{\alpha}\}_{\alpha\in A}$ cover a circle, then $|A|=2^{\aleph_0}$ (just because each line can cover at most two points on the circle). $\endgroup$
    – 喻 良
    Commented Oct 22, 2013 at 6:53
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    $\begingroup$ @LiangYu Wouldn't your nice argument also apply to a curve like the graph of $\exp$, so that the answer to 2. is also 'no'? $\endgroup$
    – Todd Trimble ♦
    Commented Oct 22, 2013 at 6:58
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    $\begingroup$ @ToddTrimble, I think you are right. $\endgroup$
    – 喻 良
    Commented Oct 22, 2013 at 7:00
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    $\begingroup$ @LiangYu: yes, this is indeed a very nice and easy geometric argument. The same circle argument works for algebraic curves since once can take the continuum of concentric circles. Would you like to write your argument as an answer so that I can close the question? $\endgroup$
    – Moishe Kohan
    Commented Oct 22, 2013 at 7:06