Shouldn’t the normal size be 2? Given, well, the name?

You’d think so, but no.
Short story is the ‘nominal’ size is the size before going into a planer to smooth the faces.
Yes, it makes little sense, like many things related to construction stuff.

It’s actually purposely milled to be under the nominal dimension because it’s designed to be 2"x4" once the plywood or sheathing is applied. It’s makes it easier to get walls that are the actual desired thickness.

Factually incorrect; the board is 2 inches by 4 inches (or whatever the marked dimension is) when rough sawn. After kiln drying and milling, it will be 1.5" thick and 3.5" wide. It still took 2 by 4 inches of the tree to make so that’s what you pay for.

Do I get the sawdust in a bag with it?

Maybe, if you ask the sawyer nicely.