OK, I will offer some guesses - but I emphasise they are guesses from a long way away.
I know exactly what you mean about the hard skin around the edges, and even the tiny blisters. But what causes it - that's a little harder. The 'burning' sensation however is very good clue: it means there is far too much friction inside your skin, usually between the hard surface layer and the very live tissure underneath that. Your skin is howling in protest.
I suspect that the footbeds in your shoes have a sort of heel cup arrangement? This is sometimes done to TRY to control the sideways movement of your heel. It's a new invention, and there is no reason to think it's a good one. In fact, there's a lot of reason to think it is bad if the bumps are much more than tiny. For the last few hundred years footbeds were always flat, and one did not have this problem.
If the footbed has edges sticking up, and if your heel problem is serious, I would take a razor blade and trim off all the bits of the footbed which are sticking up. Make the top surface dead flat. What I can say with some confidence is that a flat footbed is never a problem.
Outer edge of the balls of my feet and of my big toes
Same story. But whether you could have ridges at the edges of the footbed here is debatable. If so, flatten them. If not then I suspect that your shoes are a bit too narrow for your feet. The skin at the edges is being squashed up and rubbed. This is very common! You need wider shoes.
Few shop assistants would suggest you buy a shoe which is too 'small'. But precious few of them ever stop to see whether the shoes they are holding (selling) are too narrow for you. Many do not even know about foot width. Sure, XYZ shoes fitted Joe just wonderfully, but that does NOT mean they will fit YOU.
Measure your feet for size AND width (with socks on) on a Brannock Device and never EVER buy shoes smaller than that width! We usually go up a size (length) and stay above the measured width. We have zero foot problems.
Ingrown toenails developing on the inner part of each big toe
Same story yet again. WHY are they growing that way? Because the big toe is spendng the whole day jammed into the next toe, so the nail has nowhere else to grow. Poor foot!
Buy some shoes which have a flat footbed and are really wide enough!
Buy some loose sandals and let your feet recover for a few weeks.