A starfighter missile barrage is actually a very, very credible threat. It almost seems overpowered, really. At the same time, Starshatter manages to be neither capship- or fighter-centric. Fighters do what they would probably do in real life - lob missiles at ships, and only get into knife fights with other fighters and bombers. Getting within the defensive fire radius of a capital ship is a good way of ending up dead.
Capships seem to follow an automated repair algorithm - I actually wonder if it's exponential. Capships which are severly damaged will be repaired more swiftly, but as the damage is reduced, the rate of repair seems to go down as well. I don't like commanding capships in most instances, so I stick to fighters. Needless to say, commanding a capship is a bit more involved. There is a need to use the engineering screens, etc., etc., and other tools which let you manage the ship's resources. Actually, now that the source code is in the open, I wonder if you couldn't assemble a game which would not be unlike some of the social capship command games, where one player would work the weapons system, the other the navigation, another engineering, etc.
However, tangent aside, yes, you can and must repair your ship. Some of that is automatic, some of it you can prioritize. Do not forget that SS, despite what I would call flaws in the campaign engine, is a dynamic game (of sorts) and takes place over hours, days, and weeks. Each mission is spaced at about eight hours, and your ship will have been altered or operated on in that timeframe. The computer may have some virtual sorties within this period as well, but it's very... simplistic in comparison to what you, the player, has to deal with.