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Summer jasmine is best pruned just after flowering, in late summer or early autumn. Winter jasmine should be pruned in spring, immediately after flowering. On both types you can cut back flowered stems to a strong sideshoot and thin out crowded, crossing or misplaced branches. You should also remove any weak or thin stems. Overgrown vines can be cut back to about 2 feet from the ground.

Has it been cool and damp lately? These conditions often interfere with a plant's fragrance. Additionally, too much fertilizer (like nitrogen) can sometimes be the cause. While it creates lush growth, the blooms and subsequent smells from them may be reduced.

I think that there would be enough light in this location to keep these plants happy. You should be fine.

The older a plant is, the more likely it is to go into transplant shock when moved. This article will help you try to prevent it and will explain how to treat it if it happens after it has been moved:
https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Fall is a good time as is spring for transplanting. You are looking for relative cool but not cold temperatures with plenty of water (from rainfall or from manual watering).

Crepe myrtle cuttings are usually taken in spring or fall, while jasmine cuttings on outdoor varieties are best propagated from hardwood cuttings taken in winter, or softwood cuttings from tender varieties taken in spring or summer.

Here is more information: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/how-to-propagate-crepe-myrtle-trees.htm
https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm

It may be a jasmine of some kind.

Regardless, you should be able to use the seeds to start it, but often, flowering vines like that grown from seed can take years to mature enough to bloom. You would be better off taking cuttings from the vine and rooting them. These will be mature enough to bloom in just a year or two.

This article will help with taking and rooting cuttings:
https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm