You will go further and faster on a road bike for the same effort. You go further and faster on a better (i.e. more expensive) bike. Training is not about going further and faster for the same effort, its about putting in more effort. So, will a road bike make a difference - only if you will put in more effort on the road bike than the MTB.
If the idea of road riding spins you wheels, get a road bike because if you are enjoying it the training will happen. If the idea behind a road bike is a road bike lets you go the same speed and distance for less effort, a road bike will not help you meet your training objectives, except if you enjoy doing the same speed and distance for less effort, you are more likely to keep doing it rather than stopping altogether.
(you could pretty much 'road' in the above and other adjective describing a bicycle here) 'expensive', 'fat', 'single speed', 'Down hill', 'pink'
Plenty of people ride in sub freezing temperatures, its entirely possible to do so and enjoy. More people pack their bikes away for winter because its too hard. Only you can decide which group you want to be in.