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While typically this is done in fall, you can also transplant grape hyacinth in spring but try to wait until they have bloomed and then transplant, lifting as much of the dirt around them as possible without leaving any holes in the lawn/garden (during fall, you need only lift the bulbs). Prepare the new location before lifting the bulbs and water wellafter moving.

You can plant them now or you can store them for the summer. Tulips in particular do well when stored for the summer. Do not store them in the cold though. When storing for the summer, put them in a warm, dry place. They need to be warm in the summer and cold in the winter to bloom well.

Given that these are hardy bulbs, they generally do not require lifting unless in extremely cold regions. Planting is usually performed in fall for spring blooming. That does not mean you can't go ahead and plant them now, only that you may not see any blooms until the next season. If it gets cold enough where you live to chill them in the ground, you can plant them in fall. But, if it does not get cold enough for them to get the chill time they need to flower, you will need to pre-chill them and plant them after they are pre-chilled.

Here is an article or two 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/bgen/how-to-plant-a-flower-bulb-in-your-garden-after-winter-forcing.htm

This should help: https://www.gardeningknowhow.com/ornamental/bulbs/grape-hyacinth/planting-and-care-of-grape-hyacinths.htm

These articles should help you with that: https://www.gardeningknowhow.com/ornamental/bulbs/bgen/how-to-store-bulbs-that-have-sprouted.htm, https://www.gardeningknowhow.com/ornamental/bulbs/grape-hyacinth/planting-and-care-of-grape-hyacinths.htm

I am afraid that they can only be eradicated by digging up the plants or using a broad spectrum herbicide, which would kill everything in the area. You can minimize their spread (so you don't feel you are fighting a losing battle) by removing the foliage and any seed pods. This will keep seeds from being spread and weaken the bulbs so they can't produce bulblets.