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For (1), the existence of such metric (with better properties) is proved by Frauenfelder (see Lemma 4.3.3 of the book ``J-holomorphic curves and symplectic topology'', or Frauenfelder's paper).
to Simon: the birth death is where two critical points merge to one and disappear. The 1-d manifold is still smooth and compact, although not necessarily connected; its boundary is at where g=a or g=b.
I think this is strange. If the open disk bundle is the unit disk bundle of a vector bundle (with a metric), then just take the radius 1/2 disk bundle gives a subbundle.
