I will tell you what I know, and I got this information from someone else and so far has seem to hold true.  Lil Timmy is a super rare spawn.  When he spawns, he walks in the canal district around the cathedral district starting from the south, through the cathedral area, and around to the north entrance.  Then he despawns.  He only sells one white kitten each spawn.  It is best to look for him after a server reset.