Strange, I've been traveling with a 500cc refillable water bottle forever, never had any trouble (Germany, Netherlands, East Asia). Maybe it helps that my bottle is designed to be refilled, and not just an empty bottle from the supermarket.
Yes ... but sometimes only hot water is available, and at all times the quality of water not marked 'drinking water' could in principle be suspect (it might come via a tank somewhere — this is certainly common in old UK domestic plumbing — and the tank might have dead rat in it, etc.).