Yes but it’s incomplete; does not report all changes to a task so can’t be used for true activity/event log purposes. See this post for more.
EDIT:
Oh wait, Bastien, you’re referring to project status, not task activity. Yes, my bad, you are correct!
@Martyn_Hale, as Bastien correctly points out, my answer to #1 was incorrect - you can get the project status information from Asana without having to track the timings of it yourself - here is the documentation on how to do that.