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The use of black plastic to start seeds is normally done prior to the actual seed planting - to help warm the soil faster and prevent weed growth. Leaving this down once your seeds are in may actually prevent the seeds from germinating. I would take the plastic off if the seeds are already planted. Cats shouldn't eat the seeds and may only be attracted to the newly dug earth. A light layer of mulch can help alleviate issues with birds. Here is more help with growing zinnas: https://www.gardeningknowhow.com/ornamental/flowers/zinnia/growing-zinnia-flowers.htm

Ten inches is a good size for seedling plants. Taller varieties of zinnia need support after a few weeks of growth. Fix a stake in the soil, close to the plant stem. Leave at least a couple of inches of space between the plant and the stake, so that the stem has enough area for growth. This should help alleviate the bending or falling over issues you have previously been experiencing.

This can happen if the sides of the pot heat up, causing the potting soil to dry and shrink away from the sides of the pot.

There is also the possibility that the plant is getting too dry between waterings, causing the peat in your potting soil to dry up and shrink from the sides of the pot.

This article explains the above and how you can remedy it:
http://www.practicallygardening.com/watering-dried-out-containers/#comments

How healthy do the seedlings look? How tall are they? Are there any signs of distress at all (wilting, curling, discoloration, mildew, etc.?) Excessive rain on plants promotes disease often evidenced in stunting, spots on foliage, decay on leaves, stems or fruit, wilting and, in severe cases, death of the entire plant.
https://www.gardeningknowhow.com/plant-problems/environmental/excessive-rain-on-plants.htm

For more information on the care of zinnias, please visit the following link:
https://www.gardeningknowhow.com/ornamental/flowers/zinnia/growing-zinnia-flowers.htm

I would remove the nesting material and even move the basket until the bird chooses a new location.

The only thing I can think of that may help with the blooming is adding more phosphorus, either by using a bloom plus fertilizer or bone meal.