If I insert a disc manually, the laser will move into position, the arms will adjust two times, and then the drive will do...

The other possibility is the open/close sensor that is attached to the front underside of the drive. It is a little switch held by two plastic clips and has three wires going to it. If one of the clips breaks (which occurs often), it cause the sensor to drop to an angle and the tray does not trigger the switch everytime. You would need to put electrical tape or foam under the sensor to keep it held in place.

Several possibilities exist for this. The first is that the power supply itself is bad. These have a high failure rate now because of the age. The other possibility is that the power jack or power switch is loose, and needs the solder reflowed. The solder on these boards is weak from age, most units I work on all need reflow. There are also a few capacitors on these that need replaced. This usually won't cause it to completely be dead though, most likely you would see partial static, blank screen, or color bars. Make sure the cartridge slot and cartridges are clean. Nothing more than rubbing alcohol is needed to clean them.

Start by cleaning the cartridge slot AND any the game that you are trying to play. This is to make sure that the contacts are not dirty. If the screen is still blank, verify that the channel is set properly on the TV. Power on the system a few times and see if the game is briefly shown before going blank. This could indicate a faulty power switch, usually cold solder joints, or a failing voltage regulator. If neither of these work, the TIA chip has possibly gone bad, and it is very common with these systems. The TIA chip is what controls the video output. There is no fix for a bad TIA chip other than to replace it or buy a new Atari. These are custom chips, no generic is available.