Upvote:2
Answering your two specific questions:
Thinking about a single 180-day period is unhelpful. The 180-day period is a rolling window, meaning you have to consider all possible 180-day period (the one from July 28 to January 23, the one from July 29 to January 24, etc.) and make sure the person is never present for more than 90 days in the Schengen area in any of these periods.
The 15 days left are not βcarried overβ per se and definitely would not allow her to stay longer than 90 days in one go under short-stay rules. But they do allow her to come earlier to start another 90-day stay or to make a separate 15-day stay in-between.
In practice, I often get confused, the best solution is to use a calculator. If I entered the parameters correctly, it seems your girlfriend could come back on January 9 (not coincidentally 180 - 14 days after July 28 or 15 days before January 23).