Can schnorr-signatures be used to ensure public keys are of the correct form (namely $Y= x \cdot G$)?
As long as the public key is in the elliptic curve group, there is no such thing as being &quot;of the correct form&quot; or not. For any point $P$ in the group, there exists a value $x$ such that $P=x\cdot G$. Thus, all that is necessary is to check that $P$ is indeed in the group. This is carried out by checking (1) that $P=(x,y)$ is on the elliptic curve itself (i.e., it fulfills the equation), (2) that $P$ is not the identity (i.e., the point at infinity, or zero), and (3) that $q \cdot P = 1$, where $q$ is the order of the Elliptic curve group that you are working in. (The third check is needed when the number of points on the Elliptic curve is greater than the size of the subgroup you are working in.)
A completely separate question is whether or not the public-key owner can prove that he knows the associated private-key. That is, do they actually know $x$ such that $P=x\cdot G$. This can be done using a zero-knowledge proof, but actually a Schnorr signature is exactly a zero-knowledge proof of this fact. Indeed, providing an ECDSA signature would also work. However, you need to make sure that this can't be used to extract a meaningful signature that they never meant to sign...