1-2. R. Samson Raphael Hirsch (Genesis 10:10) explains that מלך refers to someone who leads by example, and provides personal and ethical guidance. מושל refers to someone who issues rulings and declarations, and does not lead by example.
Malbim (Genesis 37:8) explains that מלך refers to someone who rules with the consent of his subjects, while a מושל rules against others' will.
3 רדה is used to refer to rulership I Kings (5:4). It particularly connotes ruling through enforcement and infliction of blows. See Isaiah (14:6) and Psalms (110:2).
4 שלט also refers to a position of authority. See Genesis (42:6). It seems to connote control if not actual kingship, as in Ezekiel (16:30) and Psalms (119:133).
5 כבש is usually used for the act of conquering, not the state of being a ruler. See for example Genesis (1:28), Joshua (18:1), and II Samuel (8:11).
6 שרר Also refers to authority, related to שר; minister. שרים are subservient to מלכים, see Jeremiah (38:17).