Your teachers are mostly wrong. So long as you have Biology and Chemistry you won't be disadvantaged at the vast majority of universities. I know Cambridge require 3 sciences (so that's not an option for you unfortunately), and there may be a couple of others, but for the most part it won't be a problem.
From what I know, there's not really an element of prestige when it comes to medicine courses. Obviously Oxford and Cambridge raise eyebrows, but outside of them, it doesn't matter where you study (your employer is ultimately the NHS, who don't care). So just pick the uni you'd most want to study at (though I'm sure actual medicine students may be able to shed more light on this).
Yes you will be expected to have a fair bit of experience. Again, actual medicine students can tell you how much, but you'll definitely need some experience in a caring environment (hospital, care home, hospice, etc).