You might have a look at Some feedback on modelling tools, I just (very briefly) described some gizmo-snapping mouse+hotkey combos you might not have found.
The ‘Lock Rotation’ toggle in PolyEdit can be used to do some of the things you mention as a multi-step process. Eg if you want the gizmo aligned w/ a particular edge you can select that edge (in local frame mode) and the gizmo will align w/ it, then toggle on ‘Lock Rotation’, and now when you change selection (eg to a vertex or face) it will keep that alignment.