The “Package power limit notification” message appears to be a Linux kernel bug that occurs under high system load from what I can see through Google searches, but no first hand knowledge on it. Some users reported the a reset did not help, but a hard power off did (shut it down fully, then start it back up).
As for your connection stampedes causing high system load, my guess would either be that queries are getting backed up in the DB and the application is abandoning the connection and opening a new one thereby compounding the problem exponentially, or there is a cache refresh stampede. Does the application use some sort of cache, or an ORM like Hibernate with a built-in cache?
What you still need to do is to get the “show engine innodb status \G” when this starts to happen. Since you know connections shoot up, you could create a simple bash script that checks the number of connections, and once it gets over say 300 (or whatever number is well above your average connections to the point where you know it’s going bad, but before it really gets bad), have the bash script start running “show engine innodb status \G” every 5-30 seconds or something and saving the results. That could help you identify what is actually going on in the database at the time, and hopefully will help you narrow down the source.