Chances are very high that you are experiencing a botched update. My iPad Mini 2 on 8.4.1 has none of these problems (nor does my 6+). Do yourself a favor and download 8.4.1 to your computer, then restore your iPad from iTunes. After restoring try loading your backup and see if things are better, but if not your backup could be hosed, so start from scratch with another restore and no backup. The cause of what you're seeing is not the iOS itself, something is corrupted.
Quite interesting, I would have imagined a botched update would pretty much result in a non booting iPad, but it is indeed a good idea to try this out. I will do a bit later today. Thanks!