Regarding kaddish, it is asur to even think in divrei torah during kaddish, let alone to say tehilim. Once kaddish starts, you stop what you are doing.
What is halachically considered kaddish? My rebbeim have told me that it ends after daamiran bialma viemru amen. After that, the tiskabel tzloshon, is an extra part of kaddish, and at that point you can continue your davening.
As far as I know, if you gauge that without saying ashrei, you can catch up to the tzibbur, and either finish shemoneh esrei before kedusha, or start with the chazan and say it word for word with him until after kedusha.