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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 …

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.

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.

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.)

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.

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.