Does the God of Peace also make war? Yes He does. It is one of the names often used of Him. He is the LORD of Hosts (as in, armies). The world is not a paradise. Evil is real and has to be fought. Christians have to fight in this war that is not against flesh & blood.
Righteousness is simply 'rightness.' It is the same word translated just or justice. It refers to lining up to the standard. In this world every line is crooked to some degree. God alone judges perfectly. The only time He interrupts justice is for the sake of mercy, which is the basis of our hope.