Someone is unhappy if they are sad or dissatisfied, particularly with their situation.
Someone is vain if they are overly interested in their own appearance, achievements or other things about themselves.
Vain is also another word for futile, as in "I tried in vain".
You wince when you are struck with a sudden pain. When you wince, you jerk away from the pain, or pull a pained facial expression.
You are worried about something when you are thinking about it a lot, and it is keeping you from being happy. People can be worried about money, or a job interview, or how they have gone in an exam they just took.
You yearn for something when you really, really want to have it. The lizard in the picture is yearning for the freedom of being outside.