No, only high winds hurt the passing game. Snow helps. Why? Because the defense doesn't have any footing either. You can get slants and short passes all day or risk getting burned deep (which is where a lot of wind would come in).
The quality of your running back doesn't necessarily dictate how great your rushing game is. It gives you a head start. Nothing more.
The Patriots have the best offense in the history of the sport. Both running and passing. I suggest you educate yourself:
www.footballoutsiders.com.
I mean, the Patriots just smashed on the 2nd-best team in the Jaguars. Any team facing them is a significant underdog, especially at home.