Can a landlord go through my trash?

Unfortunately, yes. Once you set it out it is no longer your property and you have no reasonable expectation of privacy (There are numerous examples of this in legal case files). Many appartment complexes do this, I have seen it done ... it happens. IF he is such a D!(% as to do that ... I suggest you tell him to take it to small claims. He cannot prove that someone else digging for cans did not remove the bag and place it on the ground. Maybe your letter blew out of your mailbox and a neighbor found it and threw it in thier trash. There are many ways to get out of it if you are willing to lie. I suggest just telling them the situation and letting them know you did your best. If he continues to be a dork, tell him you are going to contact the city and complain about inaequate waste disposal points and the unsanitary conditions. Usually they just threaten and if it does not happen all the time, the problem goes away.