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My first thought was that was you are seeing is slug poo. Have you seen any there? Slugs love hostas and there are various treatments for them. Here is a good link.

https://www.gardeningknowhow.com/ornamental/foliage/hosta/hosta-pest-control.htm

If they are fully brown, then they are dead and it is safe to remove them from the plant.

This article may help: https://www.gardeningknowhow.com/plant-problems/pests/deer/deer-repellents.htm

Yes, you can cut the stalk--other than holding the blooms, it really has no purpose. Many people actually cut the stalks prior to flowering to allow more energy to go into the foliage growth.

You can normally divide these plants anytime from spring through fall. Simply dig up the clumps you want to divide and split them with a spade shovel, getting anywhere from 2-4 (or more) smaller clumps, depending on the overall size. Make sure each new clump has some roots with it and replant elsewhere. This article should help you as well: https://www.gardeningknowhow.com/garden-how-to/propagation/propgen/dividing-plants.htm

This is indicating that they are still not getting enough water. Even though they are in a big pot, in high heat, they need to be watered 2 times a day, due to the fact that water will evaporate very quickly (especially if they are clay pots).

You also need to make sure that the soil is thoroughly wet, especially if it has been allowed to dry out completely. Soiless potting mix can actually repel water if the mix is allowed to dry out and needs to be submerged in water for a few minutes to rehydrate it.