The second Kholo card will now see an older, ready, untitled Ishtarri
and move accordingly, correct?
This movement would burn the title card already on that Ishtarri and
replace it. Does this "change" in titles result in him "re-earning" the
kholo title and no longer seeing a contest, or is the title still
contested until my next untap phase?
No. The Kholo cards are title cards -- they follow the fate of the
title.
They are out of play when the title is contested.