I haven't been able to find any rigorous explanations for how the influence system works, just my general observations.
Cultural Centers don't seem to be that useful, since they only give a bonus on the planet+population IP (which is likely smaller than the civ-wide bonus, and generally insignificant). Projects that give a civ-wide boost are worth it however, as are morale resources.
The civ-wide bonus (given to all planets) seems to affect the projection of the IP field around planets. It seems to increase as you reach multiples of the nearest AI's total influence (so 200%, 300%, etc). However, I think base influence is used for the calculation of this multiplier, not including the actual bonus (or you would get a bonus from the bonus). I believe the base IP is what is displayed in the graphs.
IPs as seen in the trade screen I think is your voting power in the Galactic Senate. However, I don't know how this is calculated. I think your base IP is added to this power each turn (planets only go up to 100-200 IP normally under extreme conditions, so having millions of IPs is an order of magnitude higher than all planetary IPs combined).
Flipping planets is a roll each turn, but I see little difference between 4x influence and 100x influence for this purpose (does it max out, or is SQRT of the value?). Right now the MCC seems to be like a Loyalty +100% bonus for the opponent, but it's confusing as the numbers aren't known for the flipping calculation and the MCC is likely to be fixed at some point. Maybe it will change to double the chance on the flip rolls?
Has anyone done some research on the Influence System or have some formulas to calculate flipping?