Yup, it will probably bleed, but as you also have injected insulin, some of it will find its way into your bloodstream. I also tried to stress before, that the risk for hypo is only elevated, so you will probably notice the hypo-effect if/when it happens.
Concerning your second question I just have to say - sorry, but not really. I have no such problems myself.
What size of needles are you using? If it's too long, I can imagine it gives you problems, and that would maybe explain the burning also - you could reach muscles as you inject, instead of a little bit under your skin...
Hitting a place that bleeds a little, is on the other hand not too uncommon - happens now and then.