The short answers to the original question is no (title question), yes, and yes: basically, the CM follows the original trajectory and the conservation of momentum applies. But the question and the other answers are not directly addressing the most important element of this problem: in orbital mechanics the gravitational field is not uniform and therefore the assumptions that make the CM a useful concept no longer generally apply, although they will apply in the large vicinity of the collision.
The CM is not a useful concept when the field is not uniform, and therefore the CM does not apply for most orbital mechanics. For the collision then, the CM applies at the time of the collision, but long before and long after when the objects are in different fields, the concept of the CM is not useful. Therefore, in a collision between two planets in space, the two planets enter their pre-collision-trajectories using trajectories that have no relation to a useful CM concept. As they approach, they start to enter a more uniform field where the CM now becomes relevant, and if they stay stuck together after the collision, this immediate-pre-collision CM trajectory will be the trajectory after the collision and establish long-term trajectory of the new planet.
As for conservation of momentum, it is a useful concept here (and much more so than one commenter implied stating that it's not useful because there's an applied force). To start, think of "conservation of momentum" as the result of $F = 0$ in the equation $F = dp/dt$, so when there's no applied force, momentum is conserved. But one idea of the CM is that when there's a uniform field, the CM follows a trajectory as though it were a single mass, independent of the internal forces and dynamics, and what often happens is that the internal forces lead to dynamics that are separable from the CM trajectory. This essentially leaves two separate problems: the CM trajectory and the internal trajectories relative the CM. This is the case, for example, in collisions and explosions, where something will explode in mid-air where the motion of the parts can be calculated using conservation of momentum relative to the CM. That is, if there are no internal forces, then momentum is conserved relative to the motion of the CM. And it's also useful, for example, with masses and springs, or maybe with planets and the internal gravitational forces of the planets. This is the beauty of the CM. And this is relevant in space too, so long as the CM is relevant, which will be for a long time since it will take the particles a long time to drift out of (or have arrived from) a part of space with sufficiently different gravitational field to make the CM not relevant. When the fields are different, the CM can still be calculated, but it no longer provides anything useful.
(As is probably obvious from reading this answer, the assumption in the edit to the question, which states: "So the path of the centre of mass of the two planets is an ellipse." This is not generally the case because the gravitational fields at objects in orbit are different. Consider, for example, Mercury and Neptune, where Neptune is about $300\times$ the mass of Mercury. The CM of these will basically be Neptune's orbit with a small and fast wobble created by Mercury -- so not an ellipse. That is, for most collisions of planets, the CM is very useful in the vicinity of the collision, but not useful long before or long after the collision because at these times the planets are generally in different gravitational fields. Whereas this is the case for stones in the gravitational field of the earth: the CM will always be applicable and CM of two objects following parabolas will also be a parabola.)
