Should I throw out my make up if I got conjunctivitis?

Depends. If you or your doctor suspect that the infection might be related to your cosmetics then you should discard them. Otherwise, it is often not necessary. Although make-up is not sterile and harbors a lot of germs, you have likely been using yours for a long time prior to developing conjunctivitis and it was probably not the cause. Of course, it can't hurt to buy new make-up!
Maybe. Makeup can be colonized with bacteria. Usually there are preservatives in the makeup which help control this. If the makeup is old - say more than 2 months then replacing it can be helpful.