When the back is first closed, the camera will set the shutter to X (1/80th), so the film can be advanced quickly to the first frame without having to take it off Auto, which would otherwise try to give a long exposure in lower light or with the lens cap on. So what you're seeing is normal.
With the camera set to A, if the lens cap is on, the shutter will stay open.
Indoors, at a low ISO setting and/or with the lens set to a small opening like f/16, the exposure can be quite long. So:
Set the exposure compensation dial to 0. Set the ISO to a high number (6400 is highest). Set the lens to maximum opening. See what the meter reads. If necessary, point the camera at a bright light. You should see fluctuation in the reading, and the shutter speed should match. If the meter reads +2000, adjust the lens to a smaller opening or reduce the ISO, until you see a change in the reading.