The CD player is playing when it's turned upside down but there is no pickup standing in the right position. Has this anything to do with the laser pickup alignment? Can that be fixed or do I have to buy a new laser component?

Something sounds like it is loose. If you feel that you are good with electronics try and take it apart and see if you can fix it, or take it to someone who can take a look at it. If it works upside down, then the laser works fine.