What are these horizontal & diagonal ‘studs’ in the wall called

Chis is right in what he is saying. To further elaborate, I assume the structure has a full raftered roof. You need to be absolutely sure if any portion of the wall you want to remove is a supporting part. It is rare that a partition wall running parallel with the ceiling joists is a supporting wall, unless it spans across the center axis of the house perpendicular to the ridge board and some type of post or truss comes from a rafter to the top of that wall. When you look in the attic, are there any angle trusses attaching to the joists, most likely in the center of the building? Is there a center wall on the first floor that runs the length of the house? This is usually indicates a bearing wall. If your wall in question runs parallel with the rafters and joists it most likely is not supporting, especially if there is no joist above it. Since you will have to repair the ceiling after you remove the wall, you will need something to attach your sheet rock to, unless the existing ceiling runs the length and the upper plates of the wall you are removing are located below this ceiling. In any case, you will want to install a joist. If the removed wall is not supporting, it really doesn't matter if you install before or after. other considerations are if there is any electrical or plumbing in the wall. you gotta plan for that as well.

The answer is different between a structure supported by brick and a brick faced (brick veneer) wood structure.

It also depends if you want the door flush with the brick or inset from the brick.

The jamb extensions address the situation where the door assembly is smaller than the combined wall thicknesses

In general, doors are attached to the building structure. It sounds like your structure is the brick.

I personally like the inset look, so that would mean lining and waterproofing the brick rough opening with wood (typically 2x4s, treated). Then the door and exterior trims would attach to that.

On the interior, vapor barrier (semi permeable, like Tyvek, insulation and framing. The framing should be tied into the rough framing of the door.. Jamb extensions typically extend the door jamb out to thickness of where the drywall will be, making a natural place to cutoff the crywall.

After drywall installation, interior trims cover the small gaps between the jambs and the drywall.