From what I've seen, culture divides among every phase lane that it encounters. If you have one planet backed into a corner pumping out culture, that single lane will fill up VERY quickly, as it's only dividing up by 1. It's very unlikely that you'll be able to overwhelm that planet's culture, as your nearby stations will have at least 2 lanes to divide on.
Culture seems to be "rooted" at the originating planet, with the culture lanes pushing out in every direction. The planet info shows how much culture is being pushed down each phase lane.
When culture comes up against an enemy, they fight. The two cultures push against each other based on the strength of the culture going down that phase lane. Research topics also control this, I dunno how exactly. When a single strong culture lane hits a planet with many lines, it will start to "eat" at those other lines. That single line must be stronger than the entire planet's culture to accomplish this, however. When a hostile line is consumed, your culture will start pushing down from there.
I'm not sure how culture works when it divides up on a new planet. It may treat that planet as a new "root", and spread out culture in every way, including back down the originating phase lane. I'm not sure.
Capital ships seem to work a bit differently from fighting culture. Instead of having two lines meet and compete, the capital ships just seem to "eat" away at hostile lines. I dunno if their efficiency is divided up among the phase lanes(which would put it to waste on your lines) or if it efficiently suppresses what it sees.
I don't know about the exact rate of spread. But a phase lane's culture capacity appears to be the same as its length. Long lanes will hold a lot of culture. Hub worlds with many lanes will have an enormous total of phase lane distance to hold culture.