What should a sefardi do during tachanun when praying with ashkenazim? They don't say veyavor and when they sit they put their head on their arm. What about on Mondays/Thursdays where Sefardim sit first then stand up, but Ashkenazim stand and then sit down at the end?
Also should/can a sefardi praying with ashkenazim skip tefilah ledavid, bet yaakaov shir hamalot and ketoret?
According to Halacha Berurah 131:2, a Sepharadi davening with Ashkenazim needs to say vidui, and the 13 midot. Apparently, he should say the 13 midot with ta'amim (as he would if he were davening alone - 131:9).
If it's a Monday or Thursday, he should just say the first instance of the 13 midot. In the other times it appears in the extended tachanun, he should only say from "El melech yoshev" thru "katuv b'toratach". (Halacha Berurah 131:9)
I would guess that since it's rather dangerous to say Psalm 25 in tachanun while falling on your face (if you can't have proper kavvanah), you should not fall on your face even in an Ashkanzi shul where everyone else is. (See Halacha Berurah 131:17)
According to Halacha Berurah 132:15, you should daven as you normally would, but bow when the Ashkenazim are bowing. In practice, I've found that you'll be interrupted by kaddish in a couple of places where you wouldn't normally be interrupted in a Sepharadi shul.