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Either there is a draft that is hitting the plant, which is possible near a door; or the humidity levels are too low, which happens in heated homes a lot and with this winter being rather cold, it will be drier than it has been in past years; or when you repotted, a space was left in the soil.

Check for drafts, place the pot on a tray of pebbles filled with water to raise the humidity and tamp the soil down well to collapse any spaces that might be in the soil.

It could transfer from one plant to the other. Roses tend to be more susceptible, but a jasmine could get fungus from the rose. If you suspect your roses have it and you want to make very sure that the jasmine does not get it, start treating the jasmine with a fungicide. We are personally fond of neem oil for this. This article will tell you more about it: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

It could be a few things. Most common are too little light, too little water and too little phosphorus. Check that they get at least 5 hours of light, because they are in containers, water 1-2 times a day and give them some bone meal to help with any phosphorus deficiency. This article may help with the yellowing issue: https://www.gardeningknowhow.com/plant-problems/environmental/plant-leaves-turn-yellow.htm

Some trachelospermum species' leaves turn red as they get older, so long as you've got healthy new growth coming on, it should be okay.

Are the spots powdery in nature or simply 'bleached' looking? It could be powdery mildew or water spots. At any rate, you can treat the plant with neem oil, which should take care of any fungal or pest issues. Here is more information: https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

It will not survive the winter outside in Connecticut. You will need to bring it indoors for the winter.