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I would suspect a small rodent, such as mice or voles. The netting and the fencing would not deter them. This article will help give some tips: https://www.gardeningknowhow.com/plant-problems/pests/animals/garden-mouse-control.htm

This article gives some tips for growing big onions: https://www.gardeningknowhow.com/edible/vegetables/onion/onion-info-tips-for-growing-big-onions.htm

Yes, you can separate them and replant them like sets.

This article will help with growing larger onions: https://www.gardeningknowhow.com/edible/vegetables/onion/onion-info-tips-for-growing-big-onions.htm

Most people do cut the flower off because it will sap energy away from making the onion bulb. But, if you would like to have seeds to collect and grow, you can leave it on. If you want to collect the seeds, you do need to let the seed pod dry before collecting the seeds.

If your garden is planted in rows, you can certainly turn the soil; you might even deepen the area between the rows, to act sort of like canals, and try to drain off excess water. This is the time to think about what steps you might take after this growing season to deal with such problems in the future. Here are a couple of articles that have some food for thought: https://www.gardeningknowhow.com/plant-problems/environmental/excessive-rain-on-plants.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/yard-drainage-problems.htm