We can keep goin'
1) Yes, you should. The default ships are, by design, going to inferior to what you will be able to design at nearly any given point in the game. The AI designs its own ships. If you're not interested in the cosmetic portion of the designer, you can simply and quickly stick your functional elements directly to a hull and run with it.
2) Keeping up with the AI can be a pain, especially early in the game, as they tend to focus on military techs more than I do. Remember that ship quality is as, if not more, important than quantity, and designing your own ships will greatly help your military might. Also note that most of the time, you can still hold your own against an enemy with a moderately higher score than yours. In a related note, miniaturization is good.
3) Under Options/Interface there are some selections you can make to determine when a fleet battle screen is used, such as only when both sides have fleets.
4) Damaged ships repair themselves automatically. This happens faster if the ship is in orbit around a planet.
5) Route distance, your trade score, and your trade partners' economy/size all affect trade income, but I don't know the specific numbers.
6) Ship maintenance fees are applied to your civilization's total treasury, not a specific planets, so it doesn't matter where you build them. Note, of course, that a planet's spending is higher
while it's building a ship.
7) Nope. Shoot it or live with it. (You could always try to buy/trade for it, but it's rarely worth the cost).
8) It can be, mainly to add speed and/or range. Doubling up on constructor/freight modules will not help, though, so don't waste the money. Doubling up troop transport modules, though, can be very useful.
9) It depends. Use 'em for fodder, upgrade them (though this is often prohibitively expensive), sell/trade them to other civs, or just decomission them.