I want to be "offline" at times to everyone and while "offline", I want to be able to play something. If I play a game while in this "offline" state, will it come up under the "recent activity" section on my profile?
Yes, it will. Setting your status to "offline" is related to the Steam chat only. It won't hide your activity, it will even show you as "Now Playing" on the game's store page, achievements you earn will be shown to friends, etc.. If you are actually in offline mode, the activity and achievements will update once you go online.