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It's possible that they will recover in time with some supplemental watering.
Drought tolerant doesn't mean complete immunity to extreme water deficit, so don't hesitate to irrigate once or twice a week during a heat wave.

I don't advise moving the agave plants from the garden soil to a water vase, and especially not to cut off the roots and hope they regrow.

Growing certain species of agave with hydoponic methods and materials may be possible, but I would recommend starting with plants that have been propagated for that purpose, not your garden transplants.

The homewares display is a showpiece curiosity and not conducive to long term plant health.
The blackening on the roots is representative of healthy humates and fungal associations that are vital for absorption of nutrients and protection from soil borne plant pathogens.

It would be a good idea to go ahead and cut the affected leaves to the base. New leaves will sprout from the top of the plant. Here is more information on pruning cactus and agave. Methods will apply to both: https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/cactus-pruning-info.htm

That is terrible! There are many things you can do to deter pests. One of my favorites would be Neem oil. I would suggest applying it on a cloudy day, or at least when the sun isn't directly on it. This link will take you to more articles on organic pest control: https://www.gardeningknowhow.com/special/organic/

It is possible if it is young enough. An old one will not be able to regenerate the roots as well. It would be to same procedure for splitting pups. I will include an article for that Here: https://www.gardeningknowhow.com/ornamental/cacti-succulents/agave/growing-agave.htm

Use a potting mix of equal parts compost, good potting mix or garden soil, and either gravel, pumice, or sharp sand. Do not use peat moss; its acidity and its water-holding properties are not desirable for growing agaves.
Plant agaves so that the crown is well above the soil line and will stay that way as the soil subsides after watering. Feed with liquid fertilizer once a month during the growing season.
Provide plenty of light. Many agaves will tolerate full sun in containers, even in the desert, if kept sufficiently watered. Always water completely, and test to be sure the container is at least half dry before watering again.
https://www.gardeningknowhow.com/ornamental/cacti-succulents/agave/growing-agave-plants-in-pots.htm