I like your overall ideas for starships.
Personally, I've treated them as characters, sometimes with an equipment list.
So, a more fleshed out version of your idea (take what you want, ditch the rest):
Stringtech is obviously necessary, and measures both speed and firepower/defense. If you wanted to, you could split this capability (engines, weapons, defenses). Personally, I'd let players set it just like character generation rules, and allow them to lower the ship's abilities in some fields if appropriate (so a fast ship may have String 12/6, where it uses the 6 for combat, and the 12 for raw speed).
Just like characters, ship weapons have a maximum String /damage/ of the ship's Stringtech… normally. If you want to mount something nastier on your ship, you can. But then you have to specify it specifically (and it'll at least penalize you in ship construction somehow; probably by being visible from the outside of the ship easily, or having limited ammunition).
Nanotech is a measure of the ship's ability to sense things, and a measure of how good it's internal fabbers are (note: if you don't have SA style fabbers in your setting, you may end up using it for something else). Certainly Nanotech is a measure of the ship's internal security and nanite defenses.
High nanotech ships can morph from one form to another. Nano 8+ would easily be enough to change the ship's appearance and structure over a few weeks, or small changes quickly. At Nano 11+, the ship can make mid-combat changes. (adjust numbers to taste).
Biotech is a measure of crew life support, but it wouldn't need to be higher than 3 to keep everyone comfortable. Ships with high Bio usually use the Bio instead of String for defense, and instead of Nano for appropriate tests (growing new components, etc.)
Most ships don't have Cognitech nor Metatech, unless they're people (I.e. AI's of some kind, or Narfanator's Unbound). When they do have them, they're used the same way as a character would.
Non-AI ships don't generally have skills. Limited Ais may be represented by a few low level skills. So a ship may have autopilot at 2 and repairs at 1. Similarly, a ship might have Cog 1 to represent a very low-level AI.
Unlike most technology, ships /do/ have reserve, but it can't be used to improve skill tests; only to soak attacks. Ships that are also characters have reserve normally.
Ships /may/ have themes. The ship's crew can spend twists through them, if the players agree. Complications hit the ship. I'd recommend having even less themes when having capabilities over 10. Say 4 for max capability 11, 3 for max 12, 2 for max 13, and 1 for max 14; worst complications for all of those.
Ship Biotech, Nanotech, and Stringtech get a substantial bonus for their sheer size. I'd give them a flat +3 bonus to their stats. Or, put another way, you can build a ship with bio, nano, and/or string up to 3 higher than it's civilization's max. That said, the ship's /abilities/ (like seeing gravity waves) are limited by it's written stat-3. So a Nanotech 12 ship can see things that a nano 9 character can see, but rolls using a 12 due to it's size and massively redundant sensors. This is approximately your “Starship Scale” idea, just simplified down to character level. You might want to fudge that +3 to something else, depending on how you want your setting to feel.
Really stupidly huge ships may have even higher stats.
If you're dealing with fighter/carrier combat, I'd recommend simplifying entire squadrons of fighters (or drones) into a single “ship”. If you do this, keep track of reserve loss due to pushing themselves and that lost due to getting shot at separately, since getting shot at represents actual fighters damaged or destroyed.
Alternatively, you may want to treat fighter and drone swarms as being based on the ship's Nanotech score. So the ship takes reserve damage as fighters push themselves (burning fuel and ammo) and take hits (blowing up). And the fighter's combat abilities are based on the ship's Nanotech (instead of Stringtech). It makes a little sense, as drone coordination is more important than the size of the guns on each one.
@Pilgrim: whether a starship is part of the environment, an important object, or a character, is more a setting decision than anything else. And since Tantavalist is making a new setting, it's its choice what to do with them.