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This is Cedar-Apple Rust. It has a two part lifecycle that requires both trees to complete.

Here is an article that will help:

https://www.gardeningknowhow.com/edible/fruits/apples/cedar-apple-rust-control.htm

You will have to remove their food source. Armillaria are persistent and killing the fruiting mass is only superficial. Killing the mycelial mass is necessary. This is quite difficult, bordering impossible.

For this reason, it will have to run its course.

Removing the trees (and their roots) will remove the fungus, but if they are, indeed, Armillaria then it can be a treat in disguise! Unless you have had the mushroom professionally identified I would not consume them. If you know for certain, WITHOUT A DOUBT that they are armillaria... They make a wonderful addition to a meal.

If you can't say this with certainty, don't consume anything that you can't ID down to 100%

ESPECIALLY MUSHROOMS

Here is an article for more information:

https://www.gardeningknowhow.com/plant-problems/disease/armillaria-root-rot-control.htm

Those mushrooms are doing their best to help your lawn! They break down organic material and convert it into nutrients. The fruiting bodies are but a tiny portion of the actual mushroom. The rest of it spans, likely, your entire lawn as underground mycelium.

Trying to remove it, not only removes some of the protection of more sinister fungal infections, but the chemicals used will compound and cause the soil to become toxic, as well.

Unfortunately, without gill photos, and a good picture of the base, I am unable to identify the mushroom to its species. I can say with some certainty that it is part of the Genus, Agaricus.

If you do not want the mushrooms in the lawn because they are an eyesore, then taking away the factors that let them present fruiting bodies will be the way to control them. This will mean letting the lawn dry out and removing any shade from the area.

After all is said and done, it will cause much more harm and have no positive effects once removed.

This article will offer some more information:

https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

Unfortunately, your photo did not come through. I am unable to identify the issue. It sounds like some fungal fruiting body. That may be a good place to start.

https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

Those are Peziza and are very beneficial to breaking down soil and making it into nutrients for plants. To be sure that they don't find your plant roots a suitable snack, you should recompost the soil. Correct composting temperatures will kill off the fungus, while making the fruit bodies break down into usable nutrients for your plants. Although, using the soil won't really harm much as it is.

Picking off the fruiting bodies will be fine, but that is just a small portion of the actual mushroom. The rest is living inside the soil, throughout its entirety.

This article will help you to compost the soil to make it as nutrient rich as possible for your plants: https://www.gardeningknowhow.com/composting/basics/ultimate-beginners-guide-composting.htm

They were plucked very early, but they are more than likely some type of Coprinoid mushroom. They won't hurt anything, and are helping to break down nutrients in the soil. Keep in mind that while few in the Coprinoid family are edible... many are poisonous.