Updated 1/27/15
Here's a compilation I made of the relevant info from Mr. Brookes, along with links to his posts supporting my interpretations.
Relevant info gathered from Michael Brookes' posts so far (10:40am PST)
Influence is "tracked" system wide (does this mean displayed system wide, with individual influence per station, or calculated system wide?)
https://forums.frontier.co.uk/showthread.php?t=97577&p=1522823&viewfull=1#post1522823
Controlling stations are the largest one owned by the controlling faction
https://forums.frontier.co.uk/showthread.php?t=71327&p=1526930&viewfull=1#post1526930
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526033&viewfull=1#post1526033
In order to control a system, the controlling station must be controlled by your faction. Control is taken through civil war with the current controlling faction, which can be triggered by change in influence
https://forums.frontier.co.uk/showthread.php?t=71327&p=1526898&viewfull=1#post1526898
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526033&viewfull=1#post1526033
https://forums.frontier.co.uk/showthread.php?t=97577&p=1517496&viewfull=1#post1517496
Civil War is triggered through "influence movement differentials." Implies you want to increase your factions influence while decreasing the other's influence to trigger Civil War. Other states may interfere (boom?)
https://forums.frontier.co.uk/showthread.php?t=97577&page=33&p=1631533&viewfull=1#post1631533
Lock Downs have nothing to do with Civil War. No need to push to Lock Down to get Civil War. Might it be an interfering state?
https://forums.frontier.co.uk/showthread.php?t=97577&page=33&p=1631533&viewfull=1#post1631533
Civil wars are local to stations, and not system wide
https://forums.frontier.co.uk/showthread.php?t=97577&p=1517533&viewfull=1#post1517533
Civil wars have a cool-down period. Currently 25 days as of 1.06, and likely to drop.
https://forums.frontier.co.uk/showthread.php?t=71327&p=1517442&viewfull=1#post1517442
https://forums.frontier.co.uk/showthread.php?t=71327&page=135&p=1640144&viewfull=1#post1640144
You need to capture the smaller stations first before getting the controlling station
https://forums.frontier.co.uk/showthread.php?t=97577&p=1522843&viewfull=1#post1522843
Minor factions can only be in one state at a time (e.g. Boom, Civil Unrest, Civil War, etc), no matter how many star systems the faction is in.
https://forums.frontier.co.uk/showthread.php?t=71327&page=134&p=1639947&viewfull=1#post1639947
https://forums.frontier.co.uk/showthread.php?t=97577&page=36&p=1641690&viewfull=1#post1641690
Economic booms, like Civil War, have a cool down period.
https://forums.frontier.co.uk/showthread.php?t=97577&page=35&p=1640330&viewfull=1#post1640330
Economic booms last a maximum of about 3 weeks or when the "boom" runs out, whichever comes first. Unknown if missions may sustain an ongoing boom.
https://forums.frontier.co.uk/showthread.php?t=97577&page=35&p=1640171&viewfull=1#post1640171
Version 1.1 will not allow you to dock at stations owned by factions hostile to you. Or possibly, won't let you dock after you shoot it until you SC back. Unclear.
https://forums.frontier.co.uk/showthread.php?t=105117&page=13&p=1633276&viewfull=1#post1633276
Kills in conflict zones count towards influence IF you're cashing in combat bonds (choose factions on the Right panel)
https://forums.frontier.co.uk/showthread.php?t=71327&page=115&p=1592481&highlight=#post1592481
As of now, no way to do a peaceful take over, but it may come later (no ETA)
https://forums.frontier.co.uk/showthread.php?t=97577&p=1522909&viewfull=1#post1522909
Trading makes a big boost to the influence for the faction that owns the station. Worth it to fight for the other stations to get control of the market
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526913&viewfull=1#post1526913
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526033&viewfull=1#post1526033
Successful missions increase the influence of issuing factions (seems obvious, but nice to get documentation)
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526033&viewfull=1#post1526033
https://forums.frontier.co.uk/showthread.php?t=97577&p=1523856&viewfull=1#post1523856
The background simulation is an actual simulation, and not FD staff "faking it" (also seems obvious, but not to some)
https://forums.frontier.co.uk/showthread.php?t=97577&p=1524904&viewfull=1#post1524904
Missions and trading influence only carries influence in system for the faction you are working for - so trading in Mikunn won't help the Dukes's influence in HR 7327
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526555&viewfull=1#post1526555
Doing missions for one faction in a 2 faction system will raise its influence, and can trigger civil war to flip
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526619&viewfull=1#post1526619
Missions are due for an update
https://forums.frontier.co.uk/showthread.php?t=94899&p=1514333&viewfull=1#post1514333
Economy types of stations don't change, as they're tied to the planet type, with the possible exception of terraforming
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526668&viewfull=1#post1526668
Stations are owned by the minor faction which controls it, as is the market. There's a bug where the market description may be wrong
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526613&viewfull=1#post1526613
The market is owned by the minor faction that owns the station
https://forums.frontier.co.uk/showthread.php?t=97577&p=1524888&viewfull=1#post1524888
Stations cannot be added automatically, nor can they be updated automatically
https://forums.frontier.co.uk/showthread.php?t=97577&p=1524950&viewfull=1#post1524950
https://forums.frontier.co.uk/showthread.php?t=97577&p=1523846&viewfull=1#post1523846
https://forums.frontier.co.uk/showthread.php?t=97577&p=1522905&viewfull=1#post1522905
Population size effects the change of influence, just not directly
https://forums.frontier.co.uk/showthread.php?t=97577&p=1526005&viewfull=1#post1526005
There are 2 known bugs that are affecting the background simulation - system change queue is stalling and minor faction changes are being applied to the wrong system (no fix ETA yet) - this is why we're stuck
https://forums.frontier.co.uk/showthread.php?t=97577&p=1525078&viewfull=1#post1525078
https://forums.frontier.co.uk/showthread.php?t=97577&p=1517539&viewfull=1#post1517539
https://forums.frontier.co.uk/showthread.php?t=94899&p=1517237&viewfull=1#post1517237
https://forums.frontier.co.uk/showthread.php?t=97577&p=1517168&viewfull=1#post1517168
https://forums.frontier.co.uk/showthread.php?t=94899&p=1514363&viewfull=1#post1514363
The queue bug noted above has been fixed. The state changes to the wrong faction has been fixed
- I think I may have an example of where it didn't work as of 1/25/15
https://forums.frontier.co.uk/showthread.php?t=71327&page=111&p=1584986&highlight=#post1584986
https://forums.frontier.co.uk/showthread.php?t=103124&p=1598828&viewfull=1#post1598828
https://forums.frontier.co.uk/showthread.php?t=103141&page=17&p=1631576&highlight=#post1631576
Edit: Walt, feel free to copy/pasta this into the OP.