Do you capitalize the w in world?

No.

'World' is a common noun.

It should only be capitalized in the beginning of a sentence, in a title of a book or other piece of literature, or if it is in a company's or person's name.

It is a resounding no on this one because all planets are known WORLDs, so this would not make it a concrete noun therefore excluding it from the grammar given rule of the capital letter.

On the other hand Earth would always receive a capital as it is a concrete noun because there is only one Earth, it's the name that was given to it like Mars, Pluto, Sun etc..

Efl Teacher.