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You can prune the stalks anytime after flowering. The entire plant can be cut back in spring prior to any new growth.

This article will help: https://www.gardeningknowhow.com/ornamental/flowers/clivia/clivia-plant.htm

Yes, they are called clivia.

Clivia need a rest period in order to bloom. They are alot like amaryllis in this regard. Have you provided a rest period? This article will help you with doing that:
https://web.extension.illinois.edu/dmp/palette/110102.html

It is probably mealy bugs. This article will help: https://www.gardeningknowhow.com/plant-problems/pests/mealybugs-white-residue-on-plants-leaves.htm

Most people remove the seed pods, simply because they divert energy from the plant's next bloom towards producing seeds. If you do not want or need seeds, just cut the pod off.

With the plant being planted outdoors, you should not need any special care for them to rebloom. They will go into a dormant period all on their own and will rebloom for you next year.

I would spray about once a week for one month just to make sure you clear the pests out. I would not fertilize until you see the plant putting out new growth. Stressed plants can be easily burned by fertilizer, so it is best to wait for it to recover before doing that.