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Remove and dispose of infected leaves.
This article will help you.

https://www.gardeningknowhow.com/plant-problems/disease/learn-about-plant-rust-disease-and-rust-treatment.htm

These links will help you.
https://www.gardeningknowhow.com/ornamental/flowers/jasmine/how-to-fertilize-jasmine.htm
https://www.gardeningknowhow.com/ornamental/flowers/jasmine/star-jasmine-as-ground-cover.htm

There could be a few issues that may cause this.
This article will help you.

https://www.gardeningknowhow.com/ornamental/flowers/jasmine/jasmine-leaves-turning-yellow.htm

This is, very likely, due to the cold snap. I believe this will recover just fine, as this is a very mature plant. The only thing to do is just give it some time to recover. You may want to fertilize every spring as well. This collection of articles will give you more information on the care of these: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/

Yes, you can! This is the way it has to be done when you live outside of their natural zone. This article will give you more information on their care.

I wouldn't panic yet. Jasmine are very slow growing and it is confused! Wait it out and give your vine time.

Pruning is generally done in winter, so March was a bit late for the cut back, but it's not during very active growth.

If the soil is dry, give it water, slowly and down to 6 inches.

Do not fertilize until you see active growth.

https://www.gardeningknowhow.com/ornamental/flowers/jasmine/star-jasmine-as-ground-cover.htm