Does your lady currently ride a bike? If so what? Is this dream perhaps imposing some unrealistic expectations on her (I get this clue only from the fact that there is some pressure -- used bikes aren't an option)?
First of all, the free-hub question. Every bike sold new (with few exceptions that will not concern you) are sold with a freehub or freewheel (for your purposes, they do the same thing). At the price point you are suggesting, it is likely to be a seven-speed freewheel you will get.
Giant is one brand I know that produces a bike that can do what you want. It's the Upland model. Alloy frame, but low-level components; despite that, it should serve your needs quite well. It also comes in an Extra Small Size (15cm, IIRC). If your LBS says no, insist they find out where they can source them, or go elsewhere (a reference point: We have had several women the same size as your girlfriend on a commuter project, and we had trouble sourcing small-enough bikes for them -- the local Giant dealer in Hobart, Tasmania, came up with the model required).
IIRC, the Upland had eyelets on the rear drop-out for a rack. You should be able to get away with that plus a handlebar bag. The MTB gearing should suit her, too.
But be aware, higher quality is going to cost you more than your budget, unless you are patient and wait for the right-sized used bike to come up. The lower level components on the Upland are quite serviceable.