Probably no time limit but it depends on the policy of the hotel. Usually they do this as a (simple, no-cost) service to their guests, so a few hours or 9 hours shouldn't really matter. If you're concerned, call the hotel and ask.
That depends entirely on the hotel and city. Some countries/cities have a culture of honesty and trust, and others are a bit dodgy. Some hotels put your bag in a locked room and give you a claim ticket so that (in theory) no one else collect your bag. Other hotels just put your bag off to the side of the lobby and just let you (or, possibly, someone else) pick it up later.