Actually both.  C starbase generate influence on there own (but not a lot) and will greatly increase it for your planet.  As it degrades quickly yhe further you are from the source , winning by pure influence is hard.  Also they made it way too hare to culture flip a planet (because it was too easy and correction overshot) so you might want to wait till it is fixed before you make this a prime strategy.  It also helps if you make a planet no your border into an influence specialization (with improvements), then as you flip AI planets, do the same.

To be honest i haven't tried it but i bet it works ok.  that is a special case and as it says it only flips other planets in your influence zone.

  That ideology trait is a one-time event, applying at the moment you take it.

Yes BUT it also depends on a planets resistance.  If an AI planet is in you influence area, mouse over it It will tell you 2 things 1) your influence is x times greater that theirs, the bigger this number the faster it will flip.  2) planet is x% in rebellion when it reaches 100% it flips to you.   Some races have a high resistance to being flipped or put buildings on so it can take a long time to flip them.  And as i sad before its a bit too hard now till they adjust it back some.

that is what i usually go for .  Concentrate on the 2ed row

Don't know, next time im in the game i'll look.  You can check it out too.  ~ brings up the console,  help lists all commands   help <command> gives info on that command

i wouldn't consider a peaceful influence victory on an insane map. the victory condition is to have a pretty large % of the map in your zone of influence (some 85% or something, might be more or less depending on map size or number of starting factions). so you'll probably have to fill out lots of empty space with culture bases and put up dozens of bases around rival planets to choke their influence by waiting for a long long time. sounds like an exercise in frustration. works better on smaller maps. 

the fastest road to victory on large maps is probably the science victory. don't have to take over an insanely large map with cutlure or by force - just build up lots of research and cruise to victory.