I'm given an X velocity, a Y velocity, and an angular velocity immediately before a 2-D object hits the (horizontal) ground as well as an angle of elevation between the point of impact of the object and the object's center of mass. Neglecting friction, how could I calculate the object's post-impact X, Y and angular velocities? These velocities pertain to the object's center of mass.
You say "neglecting friction", but this is essential if the angular momentum of the ball is to have any influence on the bounce.
This is potentially a difficult calculation. Some simplifying assumptions are needed. I presume that gravity is absent - or plays an insignificant role. The direction of the spin axis is important; but since you specify that this is a 2D problem, I take it that the axis is perpendicular to the page. It is also necessary to specify what fraction (if any) of kinetic energy is lost during the collision, which is best done via the "coefficient of restitution" (COR). The moment of inertia of the ball affects any change in angular momentum; you need to decide if you are modelling a 3D problem which is constrained to move in a 2D plane, or if the objects are truly 2D. For example, the MI of a solid circular disc is $\frac12MR^2$ but the MI of a solid sphere is $\frac25MR^2$.
The best treatment which I have found is that by Rod Cross in Measurements of the Horizontal Coefficient of Restitition for a Superball and a Tennis Ball. In this paper 2 models are described. One (Garwin, eqns 1-7) is suitable for a ball with a high coefficient of restitution $(e\approx 1)$ such as a superball. The other model (Brody, eqns 8-15) is more suitable for a ball with low COR $(e \ll 1)$ like a tennis ball.
Possible effects of the spin of the ball are that the superball can bounce back towards the thrower if given some backspin, and instead of bouncing the tennis ball can roll forward if given enough topspin.