How do I clean the old mirror glass?

Use a soft dry cloth first to remove loose dust. Avoid spraying liquid glass cleaners directly onto the surface. Moisture can seep behind the frame and damage the fragile silvering. If the glass has sticky residue, lightly dampen a cloth with a mild water and vinegar mix. Wipe the surface gently and dry it immediately.

I bought an antique shaving stand but the heavy mirror keeps flopping forward and won't hold its angle. Can this loose fitting be tightened at all?

The brass friction hinges at the sides have likely worn down or loosened. They rely on constant tension to hold the heavy glass upright. You can sometimes tighten the screws holding the metal mounts. If the wood threads are stripped, a restorer can plug the holes and re-cut them so the fittings bite firmly again.

What goes in the little drawers?

These compartments held daily grooming supplies. Men stored their cut-throat razors and leather strops inside. The layout varies. Some bases have one wide drawer for long brushes, while others use separate smaller sections to keep blades away from wet items.

The wooden back of the mirror seems rough and unfinished compared to the front, is it supposed to look like a piece of scrap wood?

Yes, cabinet makers did not waste timber or time finishing surfaces that faced the wall. The backboards are usually rough-sawn pine or oak. They often show original saw marks and have a dry, oxidised surface. This coarse finish is normal practice for the period and forms an important part of the object's history.