Korea: Every time you build a science building in your capital, you get the equalivent of an RA boost.
So similar to RAs, place a queue list when your about to finish the library.
This also means you should build the regular library in your capital. If you build the Great Library do it after the regular library completes (or as an alternative build the great one in your Liberty induced settler city.)
Next, the science boost applies to all specialists, so your not locked in to just scientists (like Babylon). So fill all spots if your city is large enough.
After the standard Liberty tree first, Tradition makes a good second tree for Korea for that growth. (With the standard interupt the tree for base Rationalism right after use use PT to bulb Astronomy. With the standard free GE from either HS or Liberty.)
As to your UUS:
1. The first one is a better version of the ranged sieged weapon it replaces. Standard tactics apply there. (On defense place inside your city and in other hexes the enemy can't live long enough to attack into.)
2. As to the Turtle; personally I wouldn't built it at all; it doesn't have ocean ability that the Caravel it replaces has. And I've never yet seen the AI build an Iron Clad so you can't even build one for the achievement.
Oh, and Korea is like playing a full difficulty level lower compared to most civs, I wouldn't worry so much about re-rolling for the perfect start of on a river touching a mountain with them.