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This is a pollination issue. If the flower is not well pollinated or not pollinated at all, the fruit will not fully forma and will shrivel and die. You may need to hand pollinate. These articles will help:
https://www.gardeningknowhow.com/edible/vegetables/squash/pollinate-squash-by-hand.htm
https://www.gardeningknowhow.com/garden-how-to/beneficial/creating-a-pollinator-garden.htm
https://www.gardeningknowhow.com/garden-how-to/beneficial/insect-pollination-process.htm

You may have poor growing conditions or poor pollination.

I love the smaller size squash, they are more tender, smaller seeds. I would go ahead and pick them. If you don't wash them before you refridgerate them, they will say a bit longer in the fridge. You can also pick the flowers, stuff them with a savory mixture, batter them and fry them for a nice side dish! Yummy good! Hope that helps answer your question.

While stem split is common and usually due to extreme temperature changes and uneven watering, it never hurts to check for signs of pests, especially since they can penetrate the plant through these dried, cracked splits.

This article should help: https://www.gardeningknowhow.com/edible/vegetables/squash/squash-blossom-end-rot-causes-and-treatment.htm

This can often be attributed to poor growing conditions (which doesn't seem to fit your particular situation) or poor pollination. You may also have plants that are producing only male flowers. Sometimes you will have numerous male flowers that suddenly drop off and don't give fruit. However, you could also consider hand pollinating plants if fruiting still doesn't seem to be taking place once females come in, as in your case. Here is an article that you may find helpful: https://www.gardeningknowhow.com/edible/vegetables/squash/pollinate-squash-by-hand.htm