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It is very hard to get rid of grape hyacinth. Likely, last year, they spread seeds and very small bulblets which you would not have been able to see. They need to be handpulled to be removed. You will likely also need to do this next year as well. Make sure that they and any other grape hyacinth nearby do not go to seed. Remove the spent blossoms before they produce seed pods and this will help keep the spread of them down.

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

Here are some articles that you may find helpful: https://www.gardeningknowhow.com/ornamental/bulbs/grape-hyacinth/planting-and-care-of-grape-hyacinths.htm
https://www.gardeningknowhow.com/ornamental/bulbs/grape-hyacinth/planting-and-care-of-grape-hyacinths.htm
https://www.gardeningknowhow.com/ornamental/bulbs/bgen/how-to-store-bulbs-that-have-sprouted.htm
https://www.gardeningknowhow.com/ornamental/bulbs/bgen/storing-bulbs-in-south.htm

While it is not necessary to remove the tops, it won't hurt the plants, so go ahead.

Yes, you can plant these bulbs now without worry about storing them until fall. They require a little time to establish themselves in ground and should begin popping up next spring. These articles should help:
https://www.gardeningknowhow.com/ornamental/bulbs/grape-hyacinth/planting-and-care-of-grape-hyacinths.htm
https://www.gardeningknowhow.com/ornamental/bulbs/bgen/is-it-too-late-to-plant-bulbs-when-to-plant-bulbs.htm

Yes, these are seeds and in a naturalized setting, they will re-seed and produce addition plants. In fact, in some areas, they may even be considered somewhat invasive, popping up in many unexpected locations.--They're everywhere in my own yard. When collecting seeds, you'll need to wait for them to dry and wither. When they are ready, the pods start to turn light brown and papery. Once dry, you can plant them as you would any other seeds and once seedlings are big enough, you can transplant in the garden or container. However, they don't usually bloom the first year so you'll have to wait until the plants mature.