Thoughts from a player who, as of yesterday, finally made the jump to Noble (and who, as of this morning, played the shortest game of Civ IV he ever has when a random event resulted in his civilization being wiped out by a roving band of barbarian archers).
Stay at Immortal level. Hammurabi is a strong leader and I don't think you even broke a sweat in the Lincoln game. You're obviously ready for this level.
I'd be interested in an Inland Sea game. The second game I started this morning (after my first, humiliating defeat) is an inland sea map (on the second time I've used it) and I'm curious to see how you handle it.
Bowmen are good for cheap defense early on. I also like the suggestion of using them to pillage and prevent a neighbor from working their land in the early years to keep them in the stone age for centuries.
Given the amount of warring you'll do, the game may not last long enough to enjoy the health benefit of the UB in the late game but, if it does, that's when this UB comes in handy.
Only go for AH if your only food source in the BFC requires it. Otherwise, Hunting -> Archery. (Though I would be curious to see if you could pull off an Oracle gambit at Immortal, which might require some adjustment to this.)
And, of course, the warrior must be moved 1NE.