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They don't like to move well if there is no need, but moving it to a better location is justified. It would be a good idea to do so as soon as the heat breaks, but before cold weather.

To stop the reseeding, you would need to deadhead the flowers, which would be a chore unless they are in a place where you can mow or weed-eat them. The places where they are in large groups, you can spray with a product containing glyphosate to kill them. The ones that are among desirable plants you will need to dig up, and be sure to get the whole tuber or it will regrow.

They don't like a lot of water, especially in summer, so overwatering may kill them. You can also cover them with dirt or mulch to prevent light.

If there was a disease present, then this could definitely be related. You mentioned noticing a disease in another plant, so it is possible that it has been transmitted through contact or soil.

I really can't say much from here, other than knowing that it is an extreme stress of some kind. But your local extension can help test for disease with samples.

The bottom of this page will help you to find your closest extension:

https://www.gardeningknowhow.com/garden-how-to/info/what-is-extension-service.htm

People do plant under their trees, but it's really not the best thing to do. Trees have feeder roots within the top 12 inches of soil. Planting under the tree will disturb those roots. The best thing to do for trees is add 2 to 3 inches of mulch out to the dripline. Also, whatever you plant will be competing with the tree for water and nutrients.

If you really want to plant, be very careful and try not to dig very deeply. Here is more:

https://www.gardeningknowhow.com/ornamental/flowers/fgen/gardening-around-tree-roots.htm

Could it be the container and not the soil? The container is dark gray.

Cyclamen can be fussy, especially after a move! Sudden drooping right after purchase is usually from stress—changes in light, temperature, or humidity can do it. Since the soil still feels damp, don’t water yet; overwatering can make things worse. Place the plant in a cool, bright spot away from direct sunlight and heat sources. If the roots aren’t rotting, it might perk back up once it adjusts. You can also try misting lightly for humidity, but give it a few days before panicking—it’s likely just settling into its new home.

https://www.gardeningknowhow.com/houseplants/cyclamen/cyclamen-care.htm