Dear Rabbi, If one arrives for Mincha and the tzibur have finished ashrie and are just starting kadish, should one first say ashrie and then begin Shemoneh Esrei, or should he start Shemoneh Esrei with the tzibur and say ashrie afterwards?

Shalom, Thank you for your question. The case you are asking about (when to say Ashray if one came late to mincha) is mentioned explicitly in the Mishna Brurah (108, 14) where he writes "Someone who came late to synagogue and found the congregation already praying mincha [saying the Amidah], should pray with them, and say Ashray afterwards. The Magan Avraham writes in the name of the Zohar, that even in such a situation, one should not say Ashray (afterwards) as an obligation, but rather as someone who is reciting words of Torah". Blessings.