The connotation between "don't" and "won't" is a bit different though. "Don't" is more accusatory. "Won't" comes across as more pleading, than demanding. A slightly more neutral way to ask would be to remove the negative. "Do you acknowledge" & "do you accept"

In my part of America, all of these are correct. TbilisiTime's versions using "won't" instead of "don't" are also acceptable in my part of America too; although, I think I hear more people say "don't" here. Just like I hear more people ask "Don't you agree that he should study more?" more often here than I hear "Wouldn't you agree that he should study more?"

They're all fine. TblisiTime's changes are acceptable as well.

"They won't acknowledge the existence of a problem." <- OK The others should be: Won't you acknowledge that you caused this mess? Won't you accept that you caused this mess?