I gradually learned to cycle no hands when I was on empty, dry, straight country roads.
I only stated riding a bike in my 20s, but Mr Flinders rode as a kid and can mount while the bike is moving, the lot, so I felt I should make the effort to at least ride no hands to improve my balance. I found it to be more about confidence than balance, when it came to it. I did it bit by bit, and can now do quite a decent distance, it's nice to do it to rest the back, now and again. I thought at one point I was doing great with it, until one day when I was doing it, the other way came a bloke on a unicyle, perfectly balanced and going a good pace. Then I felt like a clumsy oaf again.
As for looking behind, when you ride a horse you learn that even turning your head slightly shifts your balance enough for the horse to feel it - you can steer a horse at low speeds just with that, with your hands folded behind your back and no reins. They make you sit on your hands in the saddle and turn so you can feel it - it's a bigger shift of weight than you'd think. It's not surprising it puts a bike out of balance when we have to turn our shoulders as well. I know some people can duck and look under their arm when cycling, but I've tried and I'm not the right shape- too small and chunky. At present, I do use a mirror, but as someone before said, I only use it to tell me there is something there, not that there is nothing there- to be sure, I turn to look before doing anything, just as I do when I drive- as even driving mirrors aren't perfect.
Looking round as a cyclist means drivers have seen you looking and become aware you may want to do something, I find some may then 'allow' you to move out, if you signal and look again.