Question about new lawn

I started a new lawn about a month ago. I live in northern Jersey and the backyard is about 50% shady. I used Scott's fall mix.

While I do have a lot of good grass...I also have a lot of bald patches (a few of the bald patches are due to my dog digging it up when I'm not paying attention to him).

I assume that being it's getting cold outside I need to wait to the spring to put more seed down? If this is correct, what month should I do this in?

Also, should I use starter grass seed for the bald patches, or should I use turf builder?

Thanks in advance.