I would think that Dean being Micheal's vessel would be the real reason they had Cas pull him out of Hell, because that is when they started the 'you have to stop the apocalypse' shpeal, which we know they didn't actually want him to do.
As for gaining influence, they never really tried very hard, Z doesn't really care what humans think so I don't think he tried to hard.
As for the control over Chuck's vision, it sounds like they induced a self-fulfilling prophecy, tell Dean the 'sword' is somewhere and when he arrives there the sword is there. As for 'losing' the 'sword' aka Dean, Sam made the anti-angel detecting hex bags. Between those and whoever dropped Dean on the plane they probably did lose him for a while. Also think, they tried to hide Dean away in the finale, and would have succeeded if Cas hadn't interfered.
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As for gaining influence, they never really tried very hard, Z doesn't really care what humans think so I don't think he tried to hard.
As for the control over Chuck's vision, it sounds like they induced a self-fulfilling prophecy, tell Dean the 'sword' is somewhere and when he arrives there the sword is there.
As for 'losing' the 'sword' aka Dean, Sam made the anti-angel detecting hex bags. Between those and whoever dropped Dean on the plane they probably did lose him for a while. Also think, they tried to hide Dean away in the finale, and would have succeeded if Cas hadn't interfered.