Does security under ROM imply exactly what?
If you can show a reduction of a security property of your protocol to the security of a hash function is the standard model, you do not need the random oracle assumption. So a proof in the ROM does not have any general (positive) meaning in the SM; hence why it is controversial. About the only general thing you can say is that some (arguably highly contrived) protocols that are secure in the ROM are insecure in the SM with a concrete hash function.

The value of a ROM proof is that it still narrows the range of attacks possible on a protocol. Unlike an unproven protocol where it could break in any number of ways, a ROM protocol will only break if the hash function does not live up our theoretic expectations of it. Further, no "natural" protocol that is secure in the ROM has ever been broken.

There is a good discussion in Katz-Lindell. Basically their conclusions is, a ROM protocol is better than no proof. A SM protocol should be preferred up to a reasonable decrease in efficiency.

A RO needs to return the same output value if queried on the same input. That is essentially why quantum randomness does not work.