First, I would widen your stance. Eyes can be deceiving, so I check mine with a quick trick. I place a ball behind each of my heels and then step away. It can be a little shocking how close they actually are compared to how I perceive them to be. You want to feel comfortable over the ball, so in my opinion there is no set distance they should be, but a good rule of thumb is your shoulder-width (a little wider for longer clubs).
Widening your stance should also help you resist that sway you have on your backswing. One of my favorite drills, because I used to sway back as well, is to place a ball underneath the middle of the outside of my back foot. It braces your back leg and forces you to coil and turn instead of sway. You should feel tension and weight in your right hamstrings at the top of your backswing and also be able to pick up your left foot. If you feel that tension and can pick up your foot, you're in a good position.
Also, on your driver video, look at how much you come out of the swing the first time. I used to do this as well and would hit the BIGGEST slices I've ever seen (like 50 yards out and 150 yards right). Just so you can see why it happens, if you put yourself at an impact position, and then stand up, you'll see how much the clubface opens. So try to stay down through the swing more. I try to feel like I'm driving my left heel into the ground on my downswing and that typically keeps me from rising up.
Hope this helps.