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Topical. I know there are good mathematical theories in which "Higgs" is used, in a geometrical sense. Would someone care to explain? To clarify, I'd like to know about Higgs bundles on Riemann surfa …

As has been said, Lie groups are our best theory encoding continuous symmetry. Lie algebra theory, which is the infinitesimal counterpart, is a theory good enough that numerous problems can be solved …

What do you want to do with an open annulus in the plane? Already open subsets of the plane may need infinitely many points added to compactify them in a sensible way.

Try http://en.wikipedia.org/wiki/Theta_characteristic . Since the product of line bundles is tensor product, "square root" is in the same sense. So your comment is correct.

Talking about the arithmetic genus (http://en.wikipedia.org/wiki/Arithmetic_genus), it's the alternating sum of Hodge numbers all of which are 0. So, in short, yes.

Putting a name to the space, it's a complex Stiefel manifold. See http://en.wikipedia.org/wiki/Stiefel_manifold. (But I wasn't the first.)

For a different way of looking at the same issue see http://en.wikipedia.org/wiki/Pochhammer_contour . The contour is set up to have winding number 0 around any point.

I'm struck by the idea that one ought to prove first that there is a Morse function, and work out orientability and other things as corollaries.