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This plant does not require pruning, unless it spreads beyond where you want it- in that case you could cut it back. Otherwise, the foliage and seed heads do not require cutting back.

Yes, but choose a variety that perhaps stays smaller.
Sea Holly will need consistent moist soil in containers. They have a long tap root, so they will not want to be disturbed.
Here is a link with care information.
https://www.gardeningknowhow.com/ornamental/flowers/sea-holly/sea-holly-flowers.htm

It is normal for them to not flower the first year. Since they are in container, and have a large taproot, then this can further set them off another year, or even a few. I would give it another year, and if you see no flowers then you may be faced with putting them in an even larger and taller container to help the taproot grow to its full potential.

This article will give you more information on the care of these: https://www.gardeningknowhow.com/ornamental/flowers/sea-holly/sea-holly-flowers.htm

Normally cuttings are kept moist but not wet. I suggest you soak the root in warm water for several hours then plant it. It is safe to plant Eryngium root now.

Eryngium zabelii 'Big Blue' should be planted in full sun and receive regular watering to establish a deep root system.

Not every plant will grow at the same rate. Give the struggling plant more moisture if needed. If you have trees or other plants in the area, it could be taking the moisture.

It is likely that the plant lacks the proper nutrients or pH. You can put a fertilizer in that is high in P (phosphorus) and K (potassium) and help facilitate flowering.

You can also spread dolomitic lime around to control pH, as sometimes When that is off, it can cause a lockout.

Here is an article with more information: https://www.gardeningknowhow.com/ornamental/flowers/sea-holly/sea-holly-flowers.htm