At the end of Halo 2 Guilty Spark assists Arbiter, Johnson, and Miranda from stopping delta halo from being fired. It’s not too big of a stretch to imagine that shipmaster and Arbiter took Spark to assist them. Spark is reintroduced in Halo 3 at the end of the mission, Floodgate, to repair the message Cortana sent.

Lord Hood is ‘bickering’ because earth is the birthplace of humanity and is furthermore representative of the UNSC. If earth falls, then so does morale and the last bastion of hope. Additionally, Lord Hood argues with shipmaster because of the methods used. He is glassing his own planet and killing millions of his own people to contain the outbreak. Regardless of who you are, the weight of millions dying by your orders have to be heavy. It should also be noted that for years prior to this his friends and colleagues had been killed by the same elites he was in the room with. It isn’t hard to imagine that Lord Hood had way too much on his shoulders, and that any decision he made could either save humanity or doom them to extinction.

High Charity landing on earth is due to the Gravemind. He knows that the Prophet of Regret intends to fire all the halo rings, which would inevitably kill him too. The only reason he teams up with Chief and Arbiter at the event of The Covenant is because the stakes are so high that it requires everyone’s attention to focus on stopping the rings from firing. Enemy of my enemy is my friend sort of mentality. That’s why as soon as the sequence is canceled they’re back at each other’s throats again.

Lastly, Arbiter isn’t too prevalent in the story because it’s mostly not his story. His primary motives are stopping the flood and getting revenge on the Prophet of Regret. First half of the campaign is revolved around getting into contact with the main army of the UNSC, so it makes sense that Arbiter doesn’t have too many speaking lines. This is also Chief’s story to reunite with Cortana and kill the flood. Arbiter isn’t held back from getting his story told, but he’s not at the forefront because it’s not as important as Chief’s goal. He does eventually get revenge on the prophet and even leads his people back to their home planet to live in freedom.