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This article will help: https://www.gardeningknowhow.com/ornamental/flowers/hibiscus/tips-for-pruning-hibiscus-plants-when-to-prune-hibiscus.htm

The branch should be cleanly pruned (cut) from the tree with sterilized shears. Then, let the wound air heal. Do not use sealant. Most often, plants can heal themselves without any problems.

It sounds like thrips. They attack the base of flowers and new growth, causing them to die. They can be difficult to see. The Sevin should help, but make sure you are focusing on the flower buds when applying. Here is more information:
https://www.gardeningknowhow.com/plant-problems/pests/insects/controlling-thrips.htm

The hibiscus flower normally blooms for only one day, opening in the morning and wilting by late afternoon. To compensate for this, the buds grow quickly and are opening all the time, sometimes producing more than a hundred blooms in a season.

Yes, there are many different types of Hibiscus. Tropicals, found in many garden centers, will only survive in the warm climates of Zone 9 and higher. They need to be brought into the house to survive in a colder zone. Hardy hibiscus, sold as hardy hibiscus, will survive to Zone 4 but I would recommend winter protection of a covering of mulch. They are very late to emerge in the Spring and should be marked with a label so they are now mistakenly dug and thrown away. Add to this all the natives and hybreds and there is a broad spectrum of hibiscus available with survival dependent on where you live. Check zones while researching. Harder hybiscus are awesome plants that produce very large flowers and are currently the rage. Many new hybreds are available - search Hardy Hibiscus Fireball or Cranberry Crush. Rose of Sharon is also classified as a hibiscus and is grown as a speciman or is great for a living hedge in warmer climates. If a hibiscus is yellow or orange, it is tropical. Just an afterthought.

If it is a hardy hibiscus, yes you can split the rootball.

As you probably know, flowers only last a day. They will fall off but are kind of unattractive until they do. Your option - let them fall or cut them off.