Does one have to stand when Kaddish is being said? Can one finish davening if kaddish was started before he finished what he had to say?
The custom among Ashkenazi Jewry is to stand during Kaddish.
If one is in the middle of davening (with the exception of being in the middle of Shemoneh Esrei) one should stop to answer the words yehei shemei rabba mevorach etc., and the amen after le’eila min kol birchata etc, ensuring that one has proper intent for the words.
Aside from these responses (and the gap between them, during which one should focus on the words of Kaddish), one may continue to pray wherever one is up to.
Sources: See Mishnah Berurah 56:8 (standing); 51:8 (stopping for Kaddish), among several places.