So, here is the crux of it. The paladin swore an oath to defend the law, then refused to do so. He ordered the death of a rival and then, in the heat of the moment, tried to murder an ally who refused to support him.
The transgression here is a failure to act with discipline. Standing by an oath, upholding the law, and refraining from wanton slaughter are all requirements of discipline and undisciplined acts are potentially violations of the code. However, circumstances could excuse him.
- When he refused to arrest the murderer, why did he refuse? If he refused out of loyalty to his ally or out of a stubborn unwillingness to oblige his political opponent (even though his opponent's demand was correct), then he's violating the code. If he refused because he saw the opponent's demand as some sort of trick outmaneuver attempt designed to steal his position, then he could be excused for taking a stand in order to show that he was not weak, although if he doesn't eventually execute the laws then he would be in violation of the code.
- When he tried to have his rival killed, why did he do it? If he did it because he was angry or felt boxed into a corner, that's a violation. If he did it in order to maintain control of the town, it could be excusable.
- When he tried to murder his own ally, was this an ally whom he had clear authority over? If he had authority over the wizard, then he could be defended for the attack on the grounds that he was maintaining discipline among his underlings. However, if the wizard was his equal and was not obliged to obey him, then it looks like he snapped and went murder-death-kill because he was upset over not receiving support. That's a failure of discipline and a code violation.
I'd say the paladin had a really bad day here. What looks bad is that his actions are all over the map. He's failed to obtain the loyalty and cooperation of his allies, who are either actively trying to bring about his downfall, causing trouble with reckless disregard for him, or are refusing to back him up. He's using violence and murder to solve problems without a clear legal justification, and he's doing it openly where everyone can see him taking the vigilante approach. He is acting like an impulsive, untrustworthy oathbreaker. He doesn't seem to have any kind of plan, and is just thoughtlessly reacting to situations in a way that doesn't demonstrate the values he's supposed to embody.
Aside from whether he's breached the code or not, I'd also wonder why the paladin would want to travel with his companions any more. One of them tried to get him thrown out of office. Another is a cursed idiot who murdered his townsfolk. The wizard he clearly has no regard for, since he tried to kill him. These people are either his enemies or are completely useless to his goals.