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Bell peppers often have a purple-black coloration at points where stems meet, but if the black color is a solid black and the area is shriveled, this would indicate the black area is the issue.

This article will help with the flower/fruit drop: https://www.gardeningknowhow.com/edible/vegetables/pepper/pepper-blossoms-falling-off.htm

Thinning the peppers will result in larger peppers but overall weight of the harvest will be the same. With vegetable annuals like peppers, you are basically deciding between lots of smaller peppers or fewer larger peppers. It is a personal choice.

This article will help with growing peppers: https://www.gardeningknowhow.com/edible/vegetables/pepper/growing-peppers.htm

I would have your soil tested. While you are adding compost and soil, it may still be deficient in one of the other macro or micro nutrients the plants need. If I had to guess, I would say it is likely potassium you are low on, but a soil test will tell you for sure.

Tomatoes and peppers both like heat, so I think as long as these collars do not interfere with the physical growth, they should not cause a problem even with heat retention.

The twine idea is a good one and "stringing up tomatoes" is pretty common for commercial tomato farms and is starting to catch on in home gardens. You just have to keep on them to make sure that they don't grow faster than you can train them up the lines.

The black color is normal for some kinds of peppers. Nothing to worry about unless you see cankers on the stems. Bell peppers often have a purple-black coloration at points where stems meet, but if the black color is a solid black and the area is shriveled, this would indicate the black are is the issue.

Here is an article or two that you may find helpful: https://www.gardeningknowhow.com/edible/vegetables/pepper/pepper-problems.htm
https://www.gardeningknowhow.com/edible/vegetables/pepper/common-pepper-plant-problems.htm

It is likely sooty mold, which is an indicator of a pest infestation. This article will help: https://www.gardeningknowhow.com/plant-problems/disease/how-to-get-rid-of-sooty-mold.htm