can a cheif minister become a prime minister in india?

Answer 1

Yes.

Chief Ministers are effectively governors while the Prime Minister the is head of state (the equivalent of a President or a Chancellor or a Premier of the federal executive branch).

Being two at the same time would be a pain in the neck and probably impossible (since you would have to be at 2 places at the same time) unless you were a chief minister of a very small country (which India is not).