1) I know the muḍāf ilayhi is in the genitive case. What about the muḍāf?
The muḍāf has no rules governing its case. As long as it has no alif lam and no tanwin it can be in any case. Its case will depend upon what it is doing in the sentence. For example, if the mudaf is a mubtada’ or a khabr it will be in the nominative (raf’) case. If it is playing a different grammatical role the sentence, it will take the case assigned to that role. In summary, the mudaf can be in any case whereas the muḍāf ilayhi must be in the genitive (jarr) case.
2) Does anything happen to the nūn of the dual or sound masculine plural if it is in the muḍāf ilayhi position?
No. The nun from the end of the dual and sound masculine plural remains if they are in the muḍāf ilayhi position. It will only drop if the dual and sound masculine plural is in the muḍāf position.