I would suggest purchasing the guide books instead of maps.The guide books have ordance survey strip maps inside so you will be okay if you don't stray too far from the route.On the Gargrave to Dufton section route fionding is quite easy and the only place you really need maps are Cross Fell and the first section to Standedge.If you get mist on these sections it can be very difficult.
I would say early June is a good time.Its not yet the tourist season and you have long hours of daylight.
Campsites at Gargrave,Malham,Horton,Hawes,Muker(slightly off route)Keld,Tanhill,Bowes,Middleton,Widdybank Farm(upper Teesdale)and Dufton.
Springers a good choice of dog.The reason I asked was due to seeing people attempt the Pennine Way with Rottweilers!