It's a balance thing. At around 8 mph (for most bikes) it gets much easier to balance, and the bike is less twitchy. At walking speeds, most bikes will feel twitchy. If you practice pedaling smoothly at low speed, work on no hands riding and other balance exercises, it'll get a lot easier.

Most dogs walk at around human walking speed, but can run at least a bit faster than a human run speed. So it's ok to go a bit faster than a walking pace. Just keep an eye on your dog to make sure she's comfortable, since she needs to build up to distances just like you do.