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Yes, It's definitely possible to move your mandevilla! The key is to do it carefully to minimize stress on the plant. Make sure to water the plant well the day before the move, dig up as much of the root ball as possible, and try to replant it quickly in its new spot. Choose a time when the weather is cooler, like early morning or late afternoon, to avoid heat stress. Your mandevilla should adjust well to its new location, especially if it's still getting a mix of sun and shade. Good luck!

The yellow flowering vines marketed as mandevilla technically are not. They are Urichtes lutea or Pentilinon luteum. It is possible for some flowers to change color. The soil acidity, climatic conditions, and other factors can play a part. Color also can mutate. These articles should help:

https://www.gardeningknowhow.com/ornamental/flowers/fgen/flowers-that-change-color.htm

https://www.gardeningknowhow.com/garden-how-to/info/what-is-plant-mutation.htm

Yes, repotting will get rid of any eggs or bugs in the soil. And cutting back mandevilla will reduce any bugs. You may still want to spray the remaining foliage with Neem oil to get rid of anything else. Here is how:

https://www.gardeningknowhow.com/ornamental/vines/mandevilla/winterizing-mandevillas.htm

https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

Yes, during the summer you can pinch back the tips to create a fuller look. In late winter or early spring, you can prune/shape the plant to encourage bushiness as well. You can cut this one back by about half before new growth in spring.

https://cecolusa.ucanr.edu/Master_Gardeners/Down_to_Earth_Plants_for_Colusa_County/

I had not heard of this problem with mandevillas, but I have read that same thing happens with another plant, dogbane. I suggest removing the plant and replacing it with another flowering annual. Here are bee friendly plants:

https://www.gardeningknowhow.com/garden-how-to/beneficial/attracting-bees.htm