There are many meanings but here it means "appropriate". "with (all) due respect" is just an expression to make disagreeing with someone sound politer. "due care" is just another term/expression. (or "due care and attention") I can't think of any other examples. I'd remember the expressions rather than this specific meaning.

'Proceed with all due caution,' = to go ahead carefully. 'With all due deference to,' = this is similar to 'treating someone with due respect,' although to defer to somebody means that, in the event of a disagreement you will concede to the other person.

due diligence, due regard to.., undue influence, undue pressure

It is due time to return the book to the library so you have to spare a moment to return the book in anyway.

Again, as other people have mentioned above, "due" means "appropriate" or "the right amount" or "the amount owed." "Due care" means not too little care, but it's also not too much either. So, "with all due respect" means "I'm showing you exactly as much respect as I think you should get." (which some times is not very much - making it a very tricky statement). "I exercised due care" means that I thought about how careful to be and was that careful - but I could have done more, I just didn't think it was necessary.