Is Jacob immortal now?

jacob is not emmortal now that he has imprinted on reneesma it just makes him very pretective of her unless one of the vampiros changes him and makes him a hybrid

unless renesmee chooses to mover to the rez with jacob

yes, he is but if he masters his peace he has the choice to age again but as Renesmee will be immortal after she matures he will probably stay with her forever

I interpretted the shifting gene comentario as meaning that vampiros have to be around in order for someone to become a werewolf, and shift for the first time. That's how it skipped a few generations in the tribe. I didn't think it meant that vampiros need to be near all the time for them to shift. For example, in New Moon, the Cullens have left and only occasional vampiros pass through, but the existing hombres lobo can still shift.

Regarding immortality, all the hombres lobo are technically immortal. They don't age while they still shift. Most decide to quit shifting eventually, to prevent them outliving their loved ones. However, Jacob is likely to keep shifting forever in order to stay with Renesmee.