My impression was that fulfilling Senate missions raises your popularity with them up until the point where you become too dominant and then your continued expansion tends to embitter them more and more.

Conversely, with the people, I always thought it was just expansion (conquest) that mattered. Once you have conquered "enough", the game gives you the go ahead to launch a civil war (telling you enough of the people would support you).

But to be honest, I never worried too much about it. I liked to do the Senate missions for rewards (they seldom got nasty on me) and had to expand to win the game anyway.