What is the name for the area above the stage...?

Question

where all the stage lights are hanging and all that metal is suspended (above the stage)?

Answer 1

The "Fly" or the "Grid"

The Grid is what everything, ultimately, is suspended from.

The Fly (or Flyspace or, as Anna says, Flytower) is where dropped scenery retreats  when it is not on stage.  Flys (never "Flies") refer to anything that is suspended over the stage, and is raised and lowered by a system of ropes and pulleys (the "Flyrail").

Always hold a special deference in your heart for the men and women who weight, balance, and manipulate the Flyrail. Theirs is an often overlooked, but crucial, job on the stage.

If you will permit a bit of editorializing, remember that Theatre is a collaborative effort. Far too many performers tend to forget that. Crew is working just as hard as performers are to make a successful show.

If you are asking this question as a performer, good for you. You need to know what is going on backstage. If you are asking this question as a Crew member, shame on you. You should already know this.

Answer 2

It depends on the theatre - if bars fly in and out, it's the flytower or the flys; if it's fixed to the roof, then it's the grid.  Just to add confusion, if there is a flytower, then the structure at the top of that (where all the pulleys are) is also called the grid.