I know that it is possible to define RSA over elliptic curves just as DSA and Diffie-Hellman have been. I know that it doesn't offer much of a speed advantage, but does it at least reduce the size of the keys?
No, it would not reduce the size of the keys. To break an ECRSA problem, you would factor the order; that factorization problem is not inherently different than any other factorization problem over the integers. Hence, to make that problem intractable, you'd need to make it as large as traditional RSA.