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I suspect two problems here. One is that your soil may not be draining well, which can cause root rot. It also sounds like you have some powdery mildew. I suggest doing a few treatments with neem oil, check on your soil drainage. Here's a article that should be helpful: https://www.gardeningknowhow.com/ornamental/flowers/jasmine/diseases-of-jasmine.htm

It looks like spider mite damage. They are hard to get rid of because insecticides don't kill them. You can try a blast of water or Neem oil.

https://www.gardeningknowhow.com/plant-problems/pests/insects/spider-mite-treatment.htm

Insects like spider mites, aphids and scale may have stopped by at some point before/after you rescued the plant. Keep an eye for them. Use a magnifying lens when looking for spider mites as they are as small as a period in a sentence. I would let the plant decide when it wants to replace the leaves with these lesions. When looking for pests, check during the day and night; look at the top/bottom of leaves and all branches; look at the soil and under rocks.

You do not want it to bloom without roots. You must, first, get it to grow roots, as flowering hormones within the plant will inhibit rooting hormones. Getting it to root doesn't NEED hormone, but it will lessen the risk of a failed rooting, drastically.

https://www.gardeningknowhow.com/garden-how-to/projects/rooting-plant-cuttings.htm

https://www.gardeningknowhow.com/garden-how-to/propagation/cuttings/root-stimulating-hormone.htm

It could be! Overwatering can cause suffocation. The soil needs time to dry out between waterings. If this is not possible, then biofungicides may be an option. They will thrive in wet conditions. It will lessen the impact of the wet soil, but it will not help with suffocation.

You can try to increase the drainage in the area. These articles will help:

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/what-is-well-drained-garden-soil.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/why-does-soil-need-aerating.htm

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/improving-compacted-soil.htm

Feeding it can help, as well:

https://www.gardeningknowhow.com/ornamental/flowers/jasmine/how-to-fertilize-jasmine.htm

Your jasmine most likely was overwatered and the roots succumbed to root rot. Most plants do best when soil is allowed to dry out between waterings, rather than a set schedule to water. You also can buy a moisture meter that will read the moisture level in the soil.

If you still have it, cut back on the watering. It also prefers rich soil and regular fertilizing with a high phosphate fertilizer like 10-20-10 during spring and summer.