Cheshire

Following: Right now, I imagine that Cheshire gets bored sometimes while wandering around. He might begin following people and trying to not be seen, mostly as a game; see if he can still disappear without actually disappearing. On the way he might overhear something or figure out where your character lives, but he won't be actively trying to remember this or be likely to bring it up. He'll probably also soon forget what was said.
 

Stalking: He has not had a desire or need to do any of this yet, but in Wonderland he liked to know everyone's business because of the unrest within the country. In the future he may want to follow around certain people to learn more about the situation and plans of various parties, and would be committing them to memory. If it is anything major I will still ask before hand, but otherwise he would try to come to learn about habits and schedules and histories of who he can. He won't be repeating this information in anything less than confusing references, but if he does, I will ask further permissions.

Room Crashing: Cheshire has a lovely habit of showing up unannounced, and is more than willing to try to sneak into people's rooms. He has no problem opening a closed but unlocked door, or opening a window, or anything else, just to sleep on their bed.