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Yes, you can grow these plants in Zone 7 and they should be fine. However, you should keep in mind that in Zone 7, you may have trouble getting them to chill over the winter. They need at least 7-8 weeks of temps below 45 to bloom the following year. These articles will help:
https://www.gardeningknowhow.com/ornamental/bulbs/bgen/how-to-plant-bulbs-in-the-south.htm
https://www.gardeningknowhow.com/ornamental/bulbs/bgen/how-to-plant-a-flower-bulb-in-your-garden-after-winter-forcing.htm

This article will help you with planting outside:
https://www.gardeningknowhow.com/ornamental/bulbs/bgen/how-to-plant-a-flower-bulb-in-your-garden-after-winter-forcing.htm

Unfortunately, we cannot offer suggestions on places to find seeds. I would search online, or at your local nurseries.

Not in one season. If They had been there a few years, it is possible, but they need to be in the ground for at least a full year before they produce seeds and bulblets (which you would need to grow new hyacinth).

They may have been there before but just did not do well (i.e. flower) before there was someone to care for the location. Seeds may have been dropped by birds or other animals as well.

If the other plants in your area have already bloomed, they should bloom within 2-3 weeks after being planted outside. The weather should be warm enough to encourage that. That being said, there is no guarantee.

You should really wait until fall to move them, but I have seen them moved in spring without problems. If you move them while in bloom, the blooms will be shorter lived.