Now I have a problem in understanding your question. You pick two arbitrary balls of size 1 and take subballs of them. Is it different from picking arbitrary two balls of size $\delta$?

According to "Introduction to Symplectic Topology" by McDuff and Salamon, it is not known. See Section "Blowing up and down". I don't know what was done after the book was written.

Thank you. I was assuming $M$ orientable, too. Could you let me know the reference of the exact sequence?

Oh I am sorry, you are right. My computations of CZ indices are wrong. I should have put $2k-1$ instead of $k$.

Thanks a lot. I didn't see that the image of 1 by PSS should change as $\epsilon$ varies. For CZ indices, the computation seems to be OK for me. 4 appears because it is twice the first Chern number of $S^2$, which is the difference for different cappings.