1. Yes. Each adds to the rate of culture spread.
2. Culture reduces a planet's allegiance to its governing empire. When the culture reaches zero (you'll receive a verbal warning that it's dropping first), the planet becomes independent and can be colonized. Culture also improves the shield mitigation for Advent units that fight within their own culture-spread areas, TEC regenerate antimatter faster when in culture, and Vasari receive a boost to their weapon power.
3. Culture strength is a 1-0 switch. Either you're in it, or you're not. If you are in it, the upgrades mentioned above apply.
To increase your own culture SPREAD, build more culture buildings at the edges of your empire. Note that culture spread is cut in half every time it passes through a gravity well.
To increase your culture POWER, upgrade your spread rate and you'll increase the bonuses described above.
To decrease enemy SPREAD, capital ships in a gravity well have a culture defense capability that improves as they level up. If you have a planet in danger of being taken over by enemy culture, capital ships can help (until you can build a media centre there).
There is nothing you can do to decrease enemy culture POWER.
-- Retro
