I found a room underground with a load of mossy stone, which I thought was rare, but there were also two chests containing a few little stacks of random stuff and a pig spinning in a cage (really). Is this normal?
That, sir, is a dungeon. If we weren't playing on Peaceful then that weird thing in the middle would be spawning monsters. It is called a spawner. It has a pig in it, rather than a monster, because we are on peaceful.
The chests (and the stuff in them) are your reward for finding/neutralising it!
If it weren't on peaceful, you can stop a spawner from spawning by placing torches near it after fighting your way in.
South West of the under-construction docks, South East of the partially-built railway station, North-by-NorthWest of the island which looks like a chopped-in-half christmas tree.
Yes - it's still relatively rare, but not nearly so much as before.
I think I needed two and a half deposits worth to get enough.
Not quite enough. There's a lot of naked blue sheep wandering around, but I still need at least five times more than what I've got so far. :P
(So if anyone has any Lapis Lazuli they don't need...)