"push the lift" is not physically correct in any way, IFMU. I truly hope you didn't find such logic in any aero discussion!
As Dave says, a canard is the same thing as a horizontal tail - it balances and controls the main wing. The big difference and advantage of a canard is that it can do this balancing act with an upforce - lift - instead of the downforce that a tail must use. Thus, a canard shares the lift, while a tail pushes down, making the main wing lift more - and use more power. The Wright Brothers used a canard for that reason.
No the prop cannot blow the lift anywhere, lift can't be blown away.
The collective on the X2 is a fly by wire stick that raises two swashplates, one below the transmission that pushes control rods inside the beefy mast to control the upper rotor, and one above the transmission that is rather conventional and controls the lower rotor.