A "going over" is a rather more thorough affair than "checking out" something.
I could check out the security system by opening a door and seeing if an alarm rings. If I wanted to go over the system, I would be analyzing what doors and windows were secured, how the alarm worked, who was called in event of breach, etc.
To check your e-mail is to quickly see if you received any, and from whom. To check out your e-mail is to see if it's working. Is the system down? (I don't think "check out your e-mail" or "check your e-mail out" is very common, or clear, in fact.)
"Look into" is clearly an investigation of some sort. I will look into why that happened.
Student or Learner