a. I think your friend is very smart.

Smartest is the superlative form of smart which doesn't fit in the sentence.

b. You know, you are my best friend in the world.

Good is an irregular adjective whose superlative form is best.

c. In my opinion, she more beautiful than her sister.

Most is used with the superlative form. In this sentence, you have the comparative of superiority.

d. To be honest, I don't think Mr John is the most patient teacher at school.

The more is correct in some parallel structures, but that's not the case here. You should use most with the superlative form.