The repair and rearm stage after each 3rd wave rearms your missiles (in addition to your ballistic ammo like every other wave) and respawns your wingmen if they died. It doesn't currently repair your own ship, and I'm not positive whether it's supposed to or not -- the repair might just refer to the wingman respawn.
On the subject of over-correcting... I started out that way, too, then about 3 waves in, I realized what I was doing: I was starting pulling one direction, then rocking my stick in other direction, not seeing it respond immediately, and pushing harder. The thing that's happening under the hood, though, is my initial input had the maneuvering thrusters pointed one direction, but then my follow-up input needs them pointing in another one -- so they've got to traverse that distance and fire back up. If you don't internally recognize that, your instinct is going to be "I'm still in the dead zone" or "the sensitivity is borked" and you'll respond to those reactions by increasing the magnitude of your input. Then, when the thrusters catch up, they're going to fire way harder than you wanted.
In subsequent plays, I've been trying to be more conscious about smoothly engaging and disengaging the stick, and trying to avoid doing hard combinations of multiple axis-maneuvers, and things have gone much better for me. A light touch and some patience has gone a long way, and I've really been digging the result. And then a couple maneuvering thrusters get blown up, and everything goes to ****.
The mouse is also a bit of a challenge, in that regard, because of the wonky combination of crosshair control in a large maneuvering dead zone on the same set of axes. I really hope they just can that notion.
I also had an epiphany, last night. The green reticle doesn't mean, as I initially thought, that the ITTS is actively tracking the target within your guns' articulation range. So you can't just get the green diamond and unleash and stop worrying about precise aim. Instead, the green diamond means that the ITTS is calculating lead on a per-weapon mount basis. So it's like having lead pips to put your crosshairs on, except in reverse -- instead of aiming your guns at a pip off-target that's calculated by your ITTS, your ITTS is aiming your guns off-target for you so that they'll hit where your crosshair is. So you still need to put the crosshair on the enemy target (and, if you're very precise, the part of the enemy target you want to hit) to hit.