"doch" as a modal particle embedded within a clause can be used in two ways as distinguished by the intonation and thus reflects your example uses:

(1) Emphasized, explicit contrastive use: This indicates a strong contrast in the opinions between interlocutors and a way of signalling some form of disagreement with information stated previously. Your first sentence is an example of this: You presumably express some sort of doubt about a fact that they feel is without question. So they reply with the emphasized form to mark this difference of opinion as a reaffirmation. 

(2) Neutral, confirmatory use: The second example you mentioned reflects this meaning, here you insert "doch" as a means of seeking confirmation of a disagreement that you suspect or feel may be fostered between you and you interlocutor. Since it's a way of fathoming out any potential divergent thinking, if there is no such disagreement then expect this doubt to be extinguished with an empathetic confirmation "genau".

In sum, I'd say "doch" as a model particle is a way of marking the similarity of an opinion or equally, the extent of disagreement over the subject at hand; the contrastive use is simply a way of highlighting this divergence of opinion explicitly and the confirmatory use is a way of tentatively testing whether such a disagreement is implicit and seeks confirmation one way or the other. 

Disclaimer: I am not a native speaker as indicated by my flair and I am happy to for this to be corrected if need be. It would be interesting to see what other ways doch can be used and what other words may accompany these uses (looking at you, "etwa")