Side to side movement is normal on a stock rocker set up. Are all of your push rods "in their right place", meaning that they are in the cup designed to receive them? Some times they pop off...

Here is a trick for your bleeders (though it's too late now that you've already used the vice-grips): use a 6 point deep well socket. If you use a metric one, it'll be a bit too tight (only a bit). Take a hammer and carefully drive it right onto the bleeder. NOW, use the ratchet handle and unscrew the bleeder, and take it to your parts store and buy new ones.

Since brake parts are so cheap, I'd just get a set of new slave cylinders and calipers and install them, instead of trying to get ancient parts to work.

These cars aren't difficult to work on, and they are simple technology compared to todays cars. Yes, there is a lot to understand, but if you can understand, you'll be that much further ahead in the long run! Remember, all cars were like this for the first 70+ years of cars, so many of the "older" folks do understand them, as one had too in order to use these as daily drivers back then. Since computer controls and EFI have come into play, most people are forgetting what it was like before that, and some younger people have never been exposed to this.

Hang in there, you'll get it with some patience!