Handling is great! Handling is only affected when the basket is connected to the handlebar and/or the front wheels. But when it's connected to the frame (doesn't turn with the wheels), it doesn't affect handling much. It's quite stable.
After a couple of weeks of training, they learned to stay and not jump out. But I still hold their leash in one hand so I can give them a quick yank when they fixate on another dog or skateboarder.