If I came late to Mincha should I wait to answer Kedusha and only then start my own Shmoneh Esrei?



No, rather you should start saying Shmoneh Esrei together with the chazzan as he begins Chazaras Hashatz following along word-for-word, and say kedusha out loud together with the tzibbur and then continue the quiet Shmoneh Esrei from Ata Kadosh.



The Alter Rebbe writes (109:1):

The following rules apply when one enters a shul and finds the tzibur davening Shemoneh Esreh (of minchah)… If he can begin Shemoneh Esreh and complete it before the sheliach tzibbur reaches Kedushah, he should daven [immediately]. If not, he should not daven until he has responded Amen to HaE-l HaKadosh, for this Amen is considered as equivalent to Kedushah. If he already heard Kedushah or he knows that he will hear it later, he does not have to wait.

Although according to the Alter Rebbe (ibid:2) the option of reciting Kedusha out loud in the middle of your Shmoneh Esrei is only if one בדיעבד already started at the same moment as the shliach tzibur. However, it should not be done לכתחילה (even by Shacharis). Nonetheless, many argue that nowadays where the entire Kedusha is recited with the Chazan (unlike the original custom where only the Chazan recited Nakdishach), it’s preferable to do so and begin with the Chazan, over waiting to begin Davening after Kedusha.


See also:

שו”ע מסביר כ”ק אדה”ז למה לא להתחיל חזרת שמו”ע עם הש”ץ חוץ אם אין ברירה, היות שיצטרכו להגיד “נקדישך” ביחד עם הש”ץ. אבל לדידן שבין כך אומרים תמיד נקדישך, איך פוסקים לפועל?