I would have them re-grout the tiles. Staining works best when the dies are mixed in when preparing new grout, and those stains that you apply over an already set grout can have an iffy result. Besides, you're trying to go from (almost) white to dark brown - any chip or crack for whatever reason will be an eyesore - a bright white speck on a dark brown grout. If you mix the die in with the new grout, it will all be uniform color and any cracks/chips/imperfections will be well concealed.
Sealing the grout is optional, some people have good enough result without it. But I don't know what the space is going to be used for - if there's a lot of foot traffic or else if considerable amount of water expected either through condensing or accidental spills, I would seal it. Again, a sealer that is mixed in, such as DuPont's StoneTech grout additive (
I would have them come back and discuss the grouting specifications in detail before they commence work again. The big box store has a department that deals with subcontractors (the actual installation team) and they do have quite a bit of leverage because they give the subs work.
Good luck!