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This could most likely be due to a lack of light. This is best in bright filtered light in a bright window.

Alternatively, this could be due to overwating, or underwatering. You should always make sure to keep the soil slightly moist.

This article will help you: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-indoors.htm

This is, almost always, due to lack of humidity. It may be that you will have to seal off the room and get a bigger humidifier. They do go through a dormancy, though. So this is not necessarily bad for them. Here is an article for their care: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-plants.htm

It is completely possible to cut it back pretty hard. You will just have to know when it blooms to know when to cut. Here is an article that will help further: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/trimming-jasmine-plants.htm

I always like to wait until right before it warms up. This will help on the stress, and be a healthier choice for the plant. Here is an article for Jasmine care: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/growing-jasmine-plants.htm

You can wash the plant leaves with a spray of water.

I would not be able to identify the material on the plant.

Jasmine flowers do not last much longer than a week. Are there still buds on it or have they dried and fallen off? If so, then it may be because you repotted; are not providing good sunlight; or are not watering properly.