While we can't identify the cause of spotting with certainty in the first photo, the second photo's symptoms resemble Peach Leaf Curl (which also affects relatives of peach, like cherry). Nothing can be done to manage either disease at present, since fungicide use will not cure existing disease. Multiple fungi and bacterial leaf spot pathogens can cause similar symptoms and can be difficult to differentiate. Is this a wild cherry or a cultivated fruiting cherry? (They both look like they could be our native cherries, though different species.) Wild cherry is usually vigorous enough that it will likely outgrow the damage just fine with little impact to long-term tree health, as long as the roots remain healthy.

Gypsy Moth, now called Spongy Moth, does not create silken nests, so you probably saw Eastern Tent Caterpillars instead. These native caterpillars might cause drastic defoliation if numerous but the trees they feed on recover just fine without intervention, and the caterpillars are a great source of food for birds trying to feed nestlings.

Miri

While trimming a couple leaves off of a plant with more isolated leaf spot would be fine, in this case, doing so would likely remove too many leaves that may stress the plant or slow its growth. You can clip off the worst of them if you prefer, but otherwise, it's probably best to just leave them as-is, and if the tree sheds some of them later as it continues to put out new growth, that's normal.

Miri