In US English, "case" is a more generic word, and usually isn't used to mean "suitcase." (I'm not sure about UK English, though....)
No, a case can be anything - a pencil case, a glasses case, a briefcase etc. A suitcase is still a suitcase (even though it's easier to carry suits in a separate bag). So I agree with Sainio, and I'm Australian and know lots of British people, so it's not just an American thing.
"trunk" is "sandık." These are much larger, normally made of wood, and would normally need two people to lift them :)