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Yes, you are right about the honeydew. The insect is probably scale, or less likely aphids; getting rid of them is the same. Here's an article on controlling these pests: https://www.gardeningknowhow.com/plant-problems/pests/scale-bug-how-to-control-plant-scale.htm
Washing with hose spray is a good first measure. After that, spray your control mixture (I like soap and water,) making sure to spray up onto the undersides of the leaves. Spray till the plant is dripping. You'll need to spray once a week for four weeks to kill all the insects - this is more important than the mixture you use.
Fortunately, the sticky honeydew is easy to remove with soap and water, so if you wash the toys with that, and the rug also, you should be fine. Consult the rug manufacturer's directions to make sure of the correct washing procedure.

Schefflera are by nature woody plants; the green stems that you see are very young, newly rooted cuttings that are fresh from the grower. Yours are older, so the stem tissue is taking on its natural woody appearance. If you prefer the small, bushy, green stemmed look, you can easily 1) prune your woody stems down close to the soil, to allow the stem to produce new growth; and 2) root the cuttings to make yourself a bunch of new plants. This article has more information: https://www.gardeningknowhow.com/houseplants/schefflera/schefflera-plant-care.htm

There are a few pests that typically affect Schefflera, such as aphids, spider mites and mealybugs. These are all treated with a standard insecticide. I like neem oil, myself. It is safe for people and pets, but very effective on bad bugs. Here is more information on it:
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

These plants will tolerate a heavy pruning, but if you want to play on the safe side, do not trim more than 1/3 of the plant back in one season.

If the plant is getting overcrowded in its current container or is it looks unhealthy, then repotting may be necessary and helpful. These articles will help with caring for your schefflera plant as well as tips for repotting: https://www.gardeningknowhow.com/houseplants/schefflera/schefflera-plant-care.htm

https://www.gardeningknowhow.com/houseplants/hpgen/learn-more-about-repotting-houseplants.htm

Yes. These articles will help:
https://www.gardeningknowhow.com/houseplants/hpgen/using-cuttings-and-leaf-cuttings-to-propagate-your-houseplants.htm