I think there is a slight mistake in the formulation of the question. CP^n is the homogeneous space U(n+1)/(U(n) \times U(1))=SU(n+1) /G with G= SU(n+1) \cap (U(n) \times U(1)). The right formulation of question (2) is: is the spin structure on CP^n (for odd n, there is unique spin structure on CP^n, see Charles Siegel's answer) U(n+1)-equivariant?

The answer is no, for a very elementary reason: if the spin structure were U(n+1)-equivariant, then it certainly were U(n)-equivariant,
where U(n) embeds into the product in the standard way. But the U(n)-action on CP^n has a fixed point and it is not too hard to see that the U(n)-representation
on the tangent space to that fixed point is isomorphic to the standard representation of U(n) on C^n. So if the spin structure were equivariant, then the fixed-point representation has to be spin, which is of course wrong.

You can ask the same question for spheres (is the spin structure on S^n SO(n+1)-equivariant), and the answer is again no. But the spin structure on S^n is Spin(n+1)-equivariant; likewise the spin structure on CP^n will be equivariant under the double cover of U(n).

What you can guess from these two examples is that the question has something to do with double covers (alias central extensions of your group by Z/2). Here is the precise relation: M a spin manifold, s a spin structure (viewed as a double cover of the frame bundle of M), G a topological group acting on M by diffeomorphisms. The spin structure defines a new group G' and a surjective homomorphism p:G' \to G with kernel. G' consists of pairs (f,t), f \in G and t is an isomorphism of spin structures f^* s \to s. The spin structure is equivariant under G', and it is G-equivariant iff there is q:G \to G', pq=id. If G is a simply-connected topological group, this is always the case, but otherwise not in general.

This discussion implies that the spin structure on CP^n is indeed SU(n+1)-equivariant, if it exists. Grassmannians and other homogeneous spaces can be dealt with in the same way.