Ofc they keep logs of everything. I don`t know how long they save them. If you write a ticket to a GM that you have had a Problem 2 days ago, he checks the logs and tells you whats wrong. For example if you didn`t get an achievement he will explain you why exactly you didn`t get it. He can eaven tell you which pary member messed it up.
They also log the chats. I once contactet a gm and asked him if he could find the name of a player i chattet with 3 days ago. He checked it and he could even tell me what we wrote.
Also if a player is reportet for botting, it sometimes takes days untill he is banned. They will check the logs and see what the player did.