Specifically, I'm referring to an ice age that covers most of the Northern and parts of the Southern Hemisphere. Long after it began, how would weather in regions along the Equator and Southern Hemisphere be affected? Would the weather be chaotic, still, a mixture of both? What are the ramifications of the climate there?
Well weather is based off of a few things, water evaporation, and energy. wind requires energy. It is the displacement of an entire atmosphere, believe it or not it takes energy to do this. for whatever reason your planet has minimal energy at the moment, hence its in an ice age. granted there are other sources of energy, the spinning of your planet has a great influence on the wind, so there will still be wind but due to the lack of thermal energy it will not likely become chaotic but will feel subdued, or breezy.
Back to water evaporation. If you want rain, water must evaporate, that's the first step in the water cycle. To state the obvious, If the first step doesn't occur then neither will the rest of them. inevitably you will have minimal precipitation, your oceans are frozen rather than evaporating. condensing and precipitating. however if a small portion of your equator has water that could evaporate you could hope for some small snow or hail storms as the clouds drift into the colder regions and immediately freezes again.
If your looking for violent weather, freeze the planet over and then melt all the ice in a relatively quickly time frame, that should make things interesting.