If doing Tashlumin for Shachrit during a Mincha minyan, should I first daven the Mincha Amidah and then go straight into the Tashlumin for Shachrit, or should I wait until that minyan reaches the Arvit Amidah, and thereby fulfill my Tashlumin of Shachrit with a minyan?
I can then find a later minyan to daven Arvit properly.
Furthermore, can I do Tachanun at the end of that Mincha Amidah, before Tashlumin for Arvit has been done or would the Tachanun be a hefsek?
When doing a tefilas tashlumin, it must be done immediately after saying the regular teffilah, and you should not wait for a minyan. (If there is a minyan there anyways, obviously daven with them) In your case the first S”E is for mincha, hear chazoras hashatz, say tachanun, as it is a continuation of S”E, then say ashrei again, and then the tefilas tashlumin. As a side point, you may not daven the tashlumin for shacharis after the time to daven mincha, because a tashlumin may only be said during the time of the next tefilah.
O:CH 108 -2, M:B ibid 11,12,13. Tefilah Khilshoch 1- 58 in the name of R’S. Z. Auerbach zt”l.