Others can detail this better than I can (since I haven't read the book yet) but supposedly the Master Chief is on a path to becoming yet another version of the Didact. 100,000 years in the past a previous Didact did actually ask 343 Guilty Spark what he would do if it were his choice. Now, in Halo 1, we have him musing on the conversation he had with (a / the) Didact and reaffirming his commitment to protecting the galaxy.

: 2. Why did Regret jump the gun and send his ships to Earth before the rest of
: the Covenant.

As others have said, Truth and Regret were in something of a power struggle for leadership of The Covenant. In The Cole Protocol for instance their two plans, that they each kept hidden from the other, ended up colliding and causing both plans to fail. Truth was seeding Humanity with small amounts of Covenant weapons that contained built in tracking devices. He was slowly mapping all of Human space. Regret had found a Human colony (of sorts, it was built out of connected asteroids) in what was left of an already glassed system. He had directed a group of Jackals to be friendly to the Humans in the hope of eventually obtaining a working nav database from them. In the end, after the two plans had basically cancelled each other out Truth and Regret pretended to trust the other again, but both were secretly looking for an advantage to gain complete control of The Covenant.

Regret jumped the gun because he thought he had found a new, artifact rich world he could exploit first, and he was right, he just didn't know it was Earth protected by 150 MAC stations, etc, etc.

: 3. What did Truth dig up in the Legendary ending of ODST?

Based on the end of Halo 2, Truth was happy at the end of ODST because he had succeeded in digging up a working portal to The Ark where he could complete his plan of wiping out all his enemies.

: 4. What questions did the Gravemind ask Cortana, and what answers did she
: give?

Another recommendation for Human Weakness. It is, in my opinion, the one good look we ever got at just how powerful the Gravemind was, even when pitted against something as uninfectable as an artificial intelligence. The Gravemind didn't just pose one vocal question after another to Cortana. He, with his sum knowledge of trillions of thinking beings, was able to directly interface with her programming, and was able to tempt her, and torture her in ways that could only apply to an AI.

His had a couple of big questions he wanted answers to:

1. Where was Earth. Cortana eventually answered this by being forced to send a distress signal to the Chief via floodified Covenant cruiser

2. What was the secret that she was managing to hide? This was answered in Halo 3 when the Chief rescues Cortana and she reveals she still had the Halo 04 activation index's data.

This second point was the Gravemind's downfall. He had assumed he was basically invincible, having begun to occupy the Ark. The Halo hanging above was just a big paperweight, or so he thought. He hadn't expected Cortana to be carrying the means to fire it, and knew he was in big trouble once that secret was revealed.

: 5. If blowing up Halo is enough to stop the flood from spreading, why did the
: Forerunner not just blow up all the Halos? Yay, flood gone!

The Flood was literally almost everywhere 100,000 years ago, before the Forerunner wiped the galaxy clean. The Forerunner had massive fleets, the ability to collapse stars and thus deny the Flood entire systems, but the flood had millions of ships and trillions of combat forms. Destroying one, or even all seven Halos would not have touched even a fraction of the flood present in the same systems, much less those in most every other star system in our galaxy.

The reason it worked for the Chief was that 99.999% of the Flood had been wiped out by the Halos. In 2552 the only active Flood anywhere were the ones on Halo 04. By destroying it the Chief stopped them from spreading beyond the ring. It would have never worked if the galaxy hadn't been wiped clean before hand.

: 6. Why does the Covenant hate humanity again?

Contact Harvest explains a lot of this, I recommend picking it up if you want a more detailed explanation. The basics are that The Covenant, as a form of government, only held together because all the member races had the common goal of following in the footsteps of the Forerunner. They had been promised paradise and ascension.

Truth, Mercy, and Regret learned the truth when some Jackals and Brutes made first contact with Humanity at Harvest. The Forerunner hadn't be taken to heaven, they had been killed and had gifted their inheritance to Humanity. If the rest of the Covenant found out it would have shattered so they kept it a secret, and proclaimed Humanity as an evil to be destroyed, so as to hide the evidence of the lie that the Halos equaled salvation.

Maybe not quite on a five year old's level, but for its various missteps Halo's story is still good, still fairly tightly written, and still awesome.