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Onion size is determined by the number of leaves on the plant. PLant them earlier for a better harvest. This article will help you:https://www.gardeningknowhow.com/edible/vegetables/onion/onion-info-tips-for-growing-big-onions.htm

You can either build one up, or till an area of ground to place them. You can start them from offsets, or seed. Many onions can go passed 100 days to maturity, so plant accordingly. Here is an article that will help:https://www.gardeningknowhow.com/edible/vegetables/onion/find-how-to-grow-onions-in-your-garden.htm

This is just another name for the Egyptian Walking Onion. These are really quite interesting. Here is more information on this perennial type onion: http://www.gardening.cornell.edu/homegardening/scene18b1.html

Were these, by chance, vidalia onions? In this case, the greens have a much milder taste. This is typical of sweet garden onions that are more grown for its bulb rather than the greens. I recommend Lisbon bunching onions for flavorful greens.

You might be surprised to learn that onions are, actually, hardy down to 20 degrees. You can plant them. If the weather takes a turn for the worse, and gets colder than that, then you will want to cover overnight with some hay or other mulch until the temperatures return to above 20.

Here is an article with more information: https://www.gardeningknowhow.com/edible/vegetables/onion/find-how-to-grow-onions-in-your-garden.htm

Because the rot started at the necks of the onions, it is most likely a storage disease called Neck Rot. These articles will help you:
https://www.rhs.org.uk/advice/profile?pid=747
http://ipm.ucanr.edu/PMG/r584101111.html