I cought strep throat from my cousin, and will it turn into scarlet fever like his did?

Possibly: Some strains of strep a strains carry an extra gene plasmid that controls the production of a chemical that enters the blood & causes the generalized red rash of scarlet fever. Not all strep a strains do, but if you did catch strep from someone with the rash it would likely produce the same rash in you within 24hrs of the first fever. Early rx prior to firm establishment of the germ might stop it.

Scarlet fever: Scarlet fever is due to a specific strain of the streptococcal bacteria. You are at risk for scarlet fever but early pharyngitis treatment may help you avoid it.