I am a Chinese postdoc at an American university, doing cancer research.

Between US-China, one article allows tax exemption for total compensations up to 3 years for teachers and researchers, whoes primary purpose is teaching or/and researching. Another article allows annual exemption of $5000 per year for service income of students & trainees, whoes soly purpose is education, training or obtaining special technical experience.

I came in U.S on April 2004. Can claim exemption for my compensation from 01-01/2007 to03/21/2007. I claimed for 2004(04-12), 2005 and 2006.

Can I also claim $5000 exemption as a trainee?

J1 Researcher from China is exempt from federal income tax for 3 years. So for 2007 you are not exempt.
Read: Your U.S. Tax Return: U.S. Tax Treaties for Professors, Teachers and Researchers

You will file resident tax return and will get standard deduction of $5,350.
You cannot file nonresident tax return and claim deduction of $5,000.