Can $\mathbb RP^2$ cover $\mathbb S^2$?

The fact that the sphere $\mathbb S^2$ is actually a twofold cover of the real projective plane shows that that projective plane is not simply connected (in fact the loop formed by "going around" any projective line once cannot be contracted, although going around it twice can be), while the sphere (like any universal cover) is simply connected. But any covering of a simply connected space must be simply connected (to contract a loop upstairs just project down, contract the image loop there, and follow suit above). So the real projective plane cannot cover the sphere.

Marc van Leeuwen's answer already solves the question completely, this is just another way to prove that the projective plane doesn't cover the $2$-sphere. If $p:X\rightarrow Y$ is a covering space, and $p(x_0)=p(y_0)$, then $p_*:\pi_1(X,x_0)\rightarrow\pi(Y,y_0)$ is injective ((this is because base point preserving homotopies in the base space lift to endpoint preserving homotopies in the total space.)) But the projective plane has non trivial fundamental group, and the sphere has trivial fundamental group, so no cover is possible.

Let me address your idea about ramified coverings. The way I have been taught about those, you need complex manifolds. But the projective plane is not orientable (the sphere is its orientation covering and is connected) so it is not a complex one dimensional manifold.

As a final note, remember that for a covering space with compact total space, such as is the projective plane, the fiber over each point in the base space is discrete in the total space and closed, thus finite by compactness.

Note that even the following is true:

suppose $\mathbb{R}P^2 \to X$ is a covering. Then the map is a homeomorphism. A nice argument involves multiplicativity of the euler characteristic as you adressed it. Hence $\chi(\mathbb{R}P^2) = 1$ and hence we get $k\cdot \chi(X) = 1$ where $k$ is the number of leaves of the covering (which must be finite since $\mathbb{R}P^2$ is compact). Hence $k = \chi(X) = 1$ and the claim follows.

So not only does $\mathbb{R}P^2$ not cover $S^2$, it acutally only covers itself.