Hello, my name is Kristina.
I have arrived in US in November 2008 on a fellowship with J-1 visa. In 2009 I got a postdoctoral posiiton with the same type of visa.
Additionally, I was a tax exempt for 2009 and 2010. I have already sent my tax forms, but I found out recently that I have to fill out form 8843. Is it a problem to send it separately?
I have a problem how to answer to Part II, question 8: Were you present in the United States as a teacher, trainee, or student for any part of 2 of the 6 prior calendar years (2004 through 2009)? because my I was in US, but not on my current position and paid by an international research program.
Would you help me, please?

It doesn't matter who paid you, what matters is the type of visa.
Unless you are a student, you are exempt from counting days for the SPT for 2 calendar years. That would mean 2008 and 2009 in your case. So in 2010 you start counting the days and if you are in the US more than 183 days, you are a resident alien. In that case you would have to file a 1040 type form and you do not file a 8843 anymore.

Thank you very much for the answer.
Does it have something to do with the tax exempt? According to my university accounting system I was still a non resident alien for tax purposes during 2010, my tax treaty ended in November 2010, the effective date to become an alien resident was January 2011.

Since you arrived in 2008, the non-resident alien status is for the 2 calendar year 2008 and 2009. You file a 8843 only as a non-resident alien
The tax treaty is valid for 2 years starting the date you arrive. So from November 2008 to November 2010. However, you can claim a tax treaty as a resident only, if the treaty has an "exception to the saving clause".
If that is the case you can claim the treaty by filing a form 8833.