How to see what applications do some actions when I power on my phone?
While the logcat option recommended by Ryan of course is a valid possibility, it is not the most user-friendly solution. I've got a better idea :)

Apps started automatically after boot registered for a certain event, called BOOT_COMPLETED. To do so, they need to request this permission (you find it under "more" on the playstore pages) called "Start after Boot" (or, with its technical name, BOOT_COMPLETED -- what a surprise!). So to figure out which apps could potentially be in here, you can use a permission checker. There are many of them in the Playstore, but I will mention only two of them to get you started:


PermissionDog1
RL Permissions2


 

Both let you browse your apps by permissions, and even try to indicate "potentially dangerous apps" -- nice side-effect in your case ;)