I, too, think Lee is being a bit antagonistic towards the Prez, but I think it all goes back to his speech during Baltar's trial. He sees things being done, both by Roslin and his father, that aren't the actions of people concerned about democracy or fairness in all cases. Bill Adama is the military head honcho, though, and the military isn't about freedom of thought amongst the lower ranks - they're just supposed to follow orders - so maybe he figures there isn't too much that can be done about that side of the equation.
Roslin, however, is a different story. And she seems to have a profound effect on the actions of the Admiral because of their close friendship. Maybe Lee figures if he can't change his old man, he can affect things from the government side... which is why he resigned from the military and joined the council in the first place - to effect some change.
Regarding Six knowing about Tigh's being a Cylon... uncertain if she knows for sure, but I'd say she strongly suspects from Tigh's questions about "switching off" certain feelings of guilt or pain. But then again, those same questions could be applied if Tigh was human as well. Six could think that Tigh just wants to learn how the Cylons can do it so he can ease his pain... just a possibility.
"Something, something, something, Dark Side... something, something, something, complete!"