We have done several app compat test passes of BP 5.1 and BP 6.0 on IE11. Our testing covers most of what you list above and more. All of the above along with the other tests that we do all passed just fine (some minor alignment issues were all we found). What version of BP 5.1 are you using? Is it R2 or SP3? Because all of our testing is done on either BP5.1 R2 or SP3.
Could you please provide some further detail around the issues that you are seeing? Also, you may want to log a bug with our support team that mentions all those details. I know that in the past we have had customers that found issues with BP that worked in our internal testing and it was traced back to meta tag settings for certain pages. Our support team was able to pinpoint which ones and get solutions for them.
Thanks. Ryan