I submitted some save files for the influence bugs, and some instruction on how to encounter the bugs. I took care to make sure that the bugs can be reproduced in no more than 12 turns.
The ticket number is: #JHW-661-98717
It should cover:
-Influence warping in weird ways so that strips of foreign influence will cover your planets.
-Recently conquered planets not producing enough influence to push back another civilization's influence so they don't culture flip.
-Orbiting ships culture flipping to opposing civilizations so they keep defending them.
-Orbiting ships culture flipping to your civilization so they keep defending them.
-How culture flipped orbiting ships are not really owned by the planet's owners (and how they return to their owner's control if you somehow manage to eject them).
If I missed an influence bug, let me know and I'll hunt them down.