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There are so many plants that can be propagated from cuttings, it would be impossible to list them all. Here's an article that should help get you started:
https://www.gardeningknowhow.com/garden-how-to/projects/rooting-plant-cuttings.htm

You can root cuttings any time, but the best time to take them from the parent plant is during a plant's active growth period. Here's an informative article:
https://www.gardeningknowhow.com/garden-how-to/projects/rooting-plant-cuttings.htm

https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-root-cuttings-from-various-shrubs-bushes-and-trees.htm

There are many ways, and many things you can use to do this! Here are some articles that will help:

https://www.gardeningknowhow.com/ornamental/flowers/roses/roses-from-cuttings.htm

https://www.gardeningknowhow.com/garden-how-to/info/vinegar-rooting-hormone.htm

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

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

https://www.gardeningknowhow.com/garden-how-to/propagation/cuttings/natural-rooting-methods.htm

I would wait about three more weeks to allow for more leaf and root growth. Then you can move them into taller but not necessarily wider pots to allow for more root growth. I would wait till spring to move them outside. It is getting late in the year and they are still small. You can keep them in a cool, frost free location and water sparingly till spring.

Sure! We have a few articles that you will find useful:

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

https://www.gardeningknowhow.com/houseplants/hpgen/using-cuttings-and-leaf-cuttings-to-propagate-your-houseplants.htm

Probably depends on the size and if it calluses well. Plumeria cuttings work well when they are about 1 foot long or so. The smaller (less than a foot) that they are, the more chance of problems. Not saying it is impossible but a 1” cutting mMay be difficult for example. Hopefully it calluses well and starts growing back next year. You can plant it in a pot now or not; your choice. If it still has leaves, I would pot it.