These are tough questions. The black hole FAQ part of the forum goes into them, and gives good links too. I think some of your questions are still debated, in fact. No doubt you are aware that in the frame of the pebble, it falls into the hole in short order, but your question is posed from the frame of a distant observer, and so you have to be careful to distinguish whose time you are talking about, and even more importantly, that time is a local quantity and extending one observer's sense of time to distant places can get semantically imprecise.