Really, only one thing happens in this chapter: first contact between the Time-Space Administration Bureau of the Nanoha universe, and the UNSC and Covenant of the Halo universe.
Halo being Halo, there was never any possibility of a peaceful meeting between the Bureau and the people of the Halo universe. The Covenant would probably have started shooting on principle and worried about the ashes later; the UNSC might be willing to talk, but not when the first words out of the Bureau are "we're in charge here". And the Bureau is too used to being in control of the situation, and has reached the point where they take their authority over all worlds, even those they don't directly administer, as a given. (As they demonstrate in the original season of Nanoha and/or the first movie, with Nanoha's world.) Perhaps under less strained conditions, a peaceful meeting between the UNSC and the Bureau might have been possible... but this isn't the story in which we'll find out. And so, the first and most immediate question: who wins in a fight, the Bureau cruiser Arthra or a fleet of Covenant battlecruisers?
The Arthra's only demonstrated attack capability is the nuclear option: the Arc-en-Ciel from the end of Nanoha A's. It is a massively long-ranged, wide-area attack that completely eradicates everything within its blast radius; chatter in the anime suggests that this is done through some kind of dimensional distortion. Covenant shields can't block something like that, and so the Arthra does get in a serious first strike: four enemy ships wiped away in a heartbeat.
There's only two problems. First, the Arc-en-Ciel has also been shown to require charging and/or recharging, so the Arthra can't fire again immediately.
Second, the Covenant are not simply going to surrender. Instead, they take their numerical superiority and demonstrate Halo's own advantages. On the technological side, that would be weight of fire and strength of defenses... but far more important is the superiority in command ability. The Covenant ships scatter, insuring that the Arc-en-Ciel won't kill them all at once if it does fire again. One of them does a microjump through Slipspace, hoping that the weapon is limited in its arc of fire.
Either preparation would have been enough to insure the Arthra's defeat even if the Arc-en-Ciel could have been fired in repeated succession. Taken by surprise, close in, the Arthra's shields (it does have them) are simply insufficient to handle a Covenant battlecruiser's full broadside, and it's forced to run where the Covenant can't follow simply to stay alive.
Thus the technological balance seems to be "whoever fires first wins". Bureau weapons are powerful and diverse enough to either ignore Covenant armor and shields or just break through them, but Bureau defenses and tactics can't match the battle-hardened Covenant designs.
It's a balance that rears up again in chapter 2, once the boarding action begins... but that's for next chapter's notes.