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It is just a fungus and is not harmful to your garden. This article will help you with getting rid of it: https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

Cucumbers and squash are in the curcubit family, but they cannot cross pollinate. It is like humans and monkeys. Same basic family, but not compatible. Here is an article from the Iowa Extension Service on why it can't happen: https://www.ipm.iastate.edu/ipm/hortnews/node/981

Now, there are cucumbers that look and taste as you describe. One that comes to mind is a Mexican Sour Cucumber. You may have a mislabled plant or seed in your garden that was mislabled by the company that sold it to you.

They could also be under pollinated cucumbers, meaning some of the seeds got pollinated while others did not, which can result in one side filling out, while the other side does not. This article explains more:
https://www.gardeningknowhow.com/edible/vegetables/cucumber/deformed-cucumbers.htm

They can be picked tiny or large or any size you want. When they get too big, however, they don't taste very good. Check the size in the grocery store for the size you like.

Your soil may be lacking iron. Chlorosis appears in the fruit before it appears in the leaves. This article will help:
https://www.gardeningknowhow.com/plant-problems/environmental/leaf-chlorosis-and-iron.htm

They have not been pollinated is the most likely answer. Using pesticides not only kills the nasty bugs but the good ones, too. Bees are struggling with viruses -

It sounds like you may have a bitter melon plant. If this is not what you meant to plant, it may have been mislabled when you got it. Does it look like this:
https://www.gardeningknowhow.com/edible/fruits/bitter-melons/growing-bitter-melons.htm