Why you would go with nRF51 which is 5-y-o design if you can go with nRF52? All new development including BT SIG mash and similar projects is done by Nordic only on nRF52...
As for your second question: nRF51822 is chip name (and so are nRF51422, nRF51824, nRF52832, nRF52810, nRF52840...) while nRF51 DK, nRF52 DK, nRF52840 PDK, nRF51 Dongle etc. (as well as code names like PCA10028, PCA10040, PCA10056...) are whole development boards meaning chip + PCB + all different components and connectors to be able to work out of the box (chip itself is pretty much useless until you put it to some PCB with power stage, antenna etc.)
Both nRF51422 (which is on the nRF51 DK and dongle) and NRF51822 should work with our Mesh SDK. However, as endnode suggested, nRF52 is a better platform to start your new project. Note that our nRF5 SDK now no longer supports nRF51.
