Are following statements valid for quantum gravity?

I'll try to clarify.

If we 're going to turn off locality ain't we throwing away continuity equation?

Any viable theory of quantum gravity should be consistent with the fact that classical general relativity is normally an excellent approximation — such a good approximation that we have not yet directly observed any problems with it. In particular, the conventional idea of locality (mathematically expressed using things like a metric with Lorentzian signature and hyperbolic equations of motion) should be an excellent approximation under normal conditions. Only under extreme conditions, beyond anything experiments have directly explored, do we expect those conventional ideas to break down. Exactly what fundamental concepts should replace the conventional ones is still an open question.

So we won't really be "throwing away" the conventional theories, just like we didn't throw away Newton's theory of gravity when general relativity came along. General relativity works better, but it still reduces to Newton's theory (to an excellent approximation) under most practical conditions.

If we have a non-local theory how are we going to maintain causality in our theory?

That's one of the fundamental questions that the whole quantum-gravity community is asking, or they might ask it this way: "How will quantum gravity ensure that the usual concept of local causality is recovered under the appropriate approximations?" That's one of the issues that makes quantum gravity such a difficult and mysterious subject.

...we put any theory under the label of quantum which doesn't satisfy classical mechanics.

Well, I guess "non-quantum" was a pretty poor choice of words.$^*$ I certainly didn't mean reverting to classical mechanics. I only meant we don't know yet which principles will need to be modified to make quantum gravity work without special asymptotic conditions (if that's even possible), and maybe I went too far when I suggested that even quantum theory itself might need to be modified in some not-yet-conceived way. I didn't have any specific proposal in mind, nor have I seen any such proposal that I think is even mildly compelling. I was only leaving room for possible surprises, because we don't know yet (at least I don't know) where quantum gravity research will ultimately lead.

$^*$ I edited that answer to change the wording of that part.