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Members of the onion family are prone to bolting. Here is an article or two that you may find helpful: https://www.gardeningknowhow.com/edible/vegetables/vgen/what-is-bolting-what-it-means-when-a-plant-bolts.htm
https://www.gardeningknowhow.com/edible/vegetables/onion/onion-bolting.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.

This article can help with storing suggestions: https://www.gardeningknowhow.com/edible/vegetables/onion/storing-onions.htm

Her onions have bolted. Here is more information: https://www.gardeningknowhow.com/edible/vegetables/vgen/what-is-bolting-what-it-means-when-a-plant-bolts.htm
https://www.gardeningknowhow.com/edible/vegetables/onion/onion-bolting.htm

You can plant them if there are roots. Generally, the pods house the onion seeds and are normally dried to remove and grow these seeds. Placing them in water is actually a new one on me, but still worth a try. Here is more information: https://www.gardeningknowhow.com/edible/vegetables/onion/collecting-onion-seeds.htm

It is better to let the onion tops lay over and fade on their own. Walking them down will not produce larger onions. Here are some tips for growing big onions:
https://www.gardeningknowhow.com/edible/vegetables/onion/onion-info-tips-for-growing-big-onions.htm

Here is a link with information on onion harvest.

https://www.gardeningknowhow.com/edible/vegetables/onion/harvesting-onions.htm

Yes, you can. As long as they are firm, they can be planted later. Store them in a paper bag in a cool, dry place and they should be fine to plant later.