I straightened him out
would be correct. This doesn't necessarily mean that you've done something offensive to a person, but it was probably unpleasant. What you're implying here is that you took some stern action against another person in an adversarial position and prevailed against them. This could range from correcting someone and proving your point in a debate, to maybe punching someone in the face to settle a physical altercation.
What you're implying is that you have in some way prevailed against this person.
We can straighten things out tonight
This would be implying that you're going to come to some resolution regarding a problem at a later time. Again, this could be appropriate in a wide range of contexts. You could be talking about settling the payment of a debt later tonight. You could be in a heated argument, know that you're going to see the person later tonight, and make this statement to imply a threat. You could be arguing about which sports team is better and, knowing that a game between the two teams will be on later that evening, make this statement to suggest that your team will win.
Couldn't you straighten the things out
The use of "the" in this sentence isn't really right. It should be more like
Couldn't you straighten things out?
This is just asking if it's possible to come to a more beneficial resolution to a problem. An example use:
Jim says "It looks like my wife and I are going to court."
George replies "Couldn't you two just straighten things out?"
... where George is obviously implying that Jim and his wife work things out together instead of going to court to wage an expensive legal battle against each other.