How about propaganda that can LOWER enemys available crews?

Are you saying like, if your culture is very pervailant over your opponent that you lower their fleet supply by some percentage?

I'm just confused by what you're saying, that's all.

Or how about ships not performing as effectively? And vice-versa when you're in friendly culture? I understand that each faction has special bonuses in friendly culture (eg. Vasari do more damage when in friendly culture).

The Advent already possess a form of this - increased shield mitigation when fighting under own culture.

It is also known that, in sports (soccer for example), when a team play in its home field, the odds of victory are increased, because they feel more obliged to protect their turf. 

Similarly, this effect could be translated to the game, Having the planet owner having a upper hand (a small bonus initially) by virtue of being the defendant, and having this effect increasing depending on the amount of culture.

I don't know. Each race already has its own culture bonuses, as well as the luxury of repair platforms/turrets/hangars (not to mention factories and starbases!) when defending their own gravity well. I don't think the defenders need another advantage.

I think what he means is, some sort of espionage (through research and getting a spy in place) whereby you lower your enemies fleet supply, not necessarily while in your culture.