The simplest answer to both question is that you're thinking of Newtonian physics and that some of the Newtonian laws of motion, including momentum conservation need to be generalised when working in relativistic regimes and you're specifically working with general relativity and curved space-time backgrounds as solutions to Einstein's Field Equations (albeit, a weird one) when working with something like an Alcubierre drive. They still apply, but you cannot apply them blindly.
For example, with regards to conservation of momentum, Newton's first law of motion says that a body either remains at rest or moves in a straight line with velocity unless a force acts on it but in Einstein's picture of gravity, "straight" lines aren't straight anymore for the planets moving around the Sun because the Sun's presence has created a sort of "dent" in space which cause "straight" paths to bend round the Sun in a curve. So, the planets could actually be said to be orbiting the Sun due to an inertial motion and not due to being subject to an attractive force, as Newton described it, which would already appear to violate Newton's first law of motion, (which is really conservation of momentum). However, in general relativity, the definition of momentum is generalised such that according to that definition, conservation of momentum is actually not violated but it reduces the familiar Newtonian definition when working in flat spacetime without curvature. What this law actually says in general relativity, is no longer that bodies move in straight lines and at constant velocity if a force doesn't act on them but that they follow paths called geodesics (basically paths that follow the curvature of spacetime).
Now setting aside that there's absolutely no evidence that anything like the Alcubierra drive actually exists in reality, it's not actually even a propulsion system. What it is, is a weird solution to Einstein's Field Equations that describes a space-time whereby space itself "moves" around a spherical flat region of spacetime, called a "warp bubble". This spacetime has the peculiar property that the geodesic of any body inside the this bubble (including your spaceship) would be a path where this body would look like it's accelerating when viewed from the outside and because it's following the geodesic path, it's not actually violating any conservation laws, including the conservation of momentum for similar reasons as the example I gave above. I think that the metric for the Alcubierre drive is also symmetrical, so hypothetically, if it were being generated by your spaceship, it would have to already be moving in order to give the apparent motion of the bubble a direction. Basically, you should not think about the Alcubierre drive as propulsion system that moves the spaceship through space, but rather the deforming of space in such a way that the space "moves" around the spaceship.
