Can I grow onions from a store bought onion that has green shoots coming out, and if so, how many onions will I get from it?
You can plant it but it won't make other onions. They do get a nice flower, however, and eventually would go to seed, which then you could plant to grow onions in the future.
How cold a temperature can onion seedlings and peas stand?
They can take it getting pretty cold. Normally, around 28F without any damage. This article will help:
https://www.gardeningknowhow.com/edible/vegetables/peas/how-low-of-a-temperature-can-peas-stand.htm
I have no idea how or when to plant or harvest onions. I keep trying to plant onion bulbs I bought last year ans they are growing long leaf-like stems again this year. Last year the seemed to die, and got eaten by neighborhood cats and dogs. This year I bought another bag of red onion sets. Please help.
This article will help you with that:
https://www.gardeningknowhow.com/edible/vegetables/onion/find-how-to-grow-onions-in-your-garden.htm
Will onions survive a May frost if the tops are showing above the soil?
As long as it's only a mild frost, they should be fine. Here is an article that you may find helpful: https://www.gardeningknowhow.com/plant-problems/environmental/frost-how-to-protect-your-plants.htm
My onions have seed pods on top of the stems. Some are open and some are not. How do I propagate the seeds?
This article will help you:
https://www.gardeningknowhow.com/edible/vegetables/onion/growing-onion-seed.htm
I have already planted my onion sets. I have heard not to put fertilizer close to the onions. How should I fertilize and with what? Thank you.
Fertilizer while the sets are young can burn them. Give them a little time to establish and then you can fertilize with a slightly phosphorous heavy fertilizer. Before fertilizing, make sure the plants are well watered and try to fertilize on a cloudy day. This will reduce the risk of burning the plants.
My onion plants didn't produce last year, so I never dug them up. This year they have come up, and now they are growing seed pods. My question is: will they produce onions this year, or do I need to dig them up and start over?
They may produce bulbs this year, but it may be iffy. The same lack that caused them not to produce last year likely still a factor. Try adding some bone meal to the soil to help with bulb production.