In this [paper of Borel][1] it is shown that any non-trivial word is a dominant map from $G \times G$ to $G$ whenever $G$ is a compact connected semisimple Lie group.  <strike>So the answer to your second question is "yes".</strike>  I don't recall offhand exactly what techniques are used, but I vaguely recall that one starts with the $SL_2$ case (which contains free subgroups that one can play with) and builds up from there.

EDIT: Dominance would imply surjectivity (or at least that the image is Zariski dense) in an algebraically closed field, but I didn't realise that the question is over the reals, and so Borel's result does not fully resolve the question.


  [1]: http://www.ams.org/mathscinet-getitem?mr=702738