Answer posted by Lubos Motl in the comments; I reproduce most of it here. This answer was posted in order to remove this question from the "unanswered" list.

Some (sketches of) answers to your questions, one by one:

1. Physical states have to be invariant under gauge symmetries, so all of them are singlets and there are no nontrivial representations,
2. (and 3.) The above claim may be relaxed for gauge symmetry transformations that remain nontrivial even at infinity, so this "global part" of the gauge symmetries can admit nonzero charges and produce "global symmetries" out of the local one.
3. In general relativity one generally expects that all global symmetries arise from local symmetries in this way, so only local ones are fundamentally possible
4. Conservation of electric charge (in electrodynamics) is due to global $U(1)$ symmetry.
5. Whenever an action exists, a symmetry is really a transformation (rule) that keeps the action invariant.
6. Keeping the Lagrangian invariant at each point means to have a symmetry that doesn't mix any points, a very "internal one", and e.g. diffeomorphisms and translations don't count, but it can certainly happen.
7. Non-Abelian symmetries are equally redundancies as $U(1)$ gauge symmetries but $U(1)$ gauge symmetries are easier to be gauge-fixed at the quantum level, e.g. no $bc$ ghosts are required because the functional determinants in gauge-fixing are constant etc.