In the older Civs only the marines could attack directly from the ships. So I simply forgot about it and never tried it. But suddenly I realised it is quite possible. Luckily the AI never use the trick or the coast defence would be impossible. 16-20 units suddenly emerging from the nothing and attacking the same turn will take almost any town immediately.
So is it possible? Do you use it? Did someone saw the AI to use it? Maybe besides with marines?

All units can attack directly from ships, but receive a penalty in combat for doing so. There's a promotion you can give that removes this penalty (marines start with it). The AI rarely made use of this, but it sounds like they've changed that for BTS, as people are reporting the Ai carrying out major amphibious attacks.

It was made like this to fix the irritating situation in Civ 3 where one tile islands were unconquerable till the modern age.

Yes all units can attack directly from a ship.

BTW, in BtS the AI is more than willing to launch an assualt from a ship.

You get a 25% penalty for attacking of a ship. As said before Marines start with the Amphibious promotion, so do Berserkers (the Vikings UU). Most units can eventually pick up this promotion after they get Combat II.