In a recent game where I controlled the seas and was trying to conquer the world, I had a fleet of about 10 assorted units bombarding coastal cities before I moved in.. The battleships made more noise but didn't seem to do any more damage on tha average. Are smaller ships more cost-effective for bombardment?

in my experience battleships are much better at bombardment than smaller ships.

In my experience battleships have more succes when bombarding 1 tile away compared to 2 tiles away.

Well I think that most of the smaller ships are not really more cost-effective for bombardment. But I think that destroyer is more cost-efficient. Destroyer really happens to beat the battleship in all forms, when comparing it to its cost.

i have only one point:
battleships should be able to bombard more than once!

In my experience, battleships have proven much more effective at destroying terrain improvements and city improvements than the other bombarding ships. They seem to be about the same when shooting at exposed units (although the 2-tile radius is better, obviously). I don't know if I've just had better luck than others, but my battleship bombardments seem to hit 6 or 7 times out of 10, and that's significantly better than any of the other ships.

The Battleship is nice for attacking coastal settlements, but I recently ran into a puny galley. Thinking this would be no contest the galley went blow for blow with my battleship. I did win but the battleship took on considerable damage. The Iron Clad destroyed the Frigate that it came across.

The Battleship's best use is Destroying infrastructure.
They can wipe out railroads and cut the enemy civs ability to produce anything, or move anything.

in my last game i used battleships and destroyers to bombard a secluded jap city on a small island. My destroyers were scoring on every bombardment, either destroying some improvements, population in the city or some improvements. My battleship on the other hand couldnt hit the broad side of a barn. I was persistant but it never paied off. Must have bombarded like 5 or 6 turns with the battleship with no effect. I decided to pull it off and kept it closer to the jap border to interspet any re-enforcements they would send, made more sence than leaving it where it was just wasting ammo.