No, not waterproof. They breathe and evaporate the moisture, so better than waterproof - except if they're sitting for long periods of time as these kinds of nappies are prone to pressure leaks.
They dry pretty quickly as they're not absorbent.
You wash them, then lanolise. You need to redo it when you wash as you'd be stripping out the lanolin during washing. machine washing is ok for some covers, it depends. I have DIY ones that woudl probably fall apart in the machine, for eg, but some woven ones you can buy would be fine in the machine. Hand washing is easy anyway.