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Leave them. I would not trim the leaves, as the plant will use the leaves to create the energy that it will then store in the roots, which results in a larger root.

Horseradish is pretty easy and forgiving. If you plant a piece of the root in some soil - at least 3-6 inches down, it should start to grow. Be careful where you plant it though, as it can become invasive.

Once the leaves turn brow, you can remove them from the plant. They are dead and can no longer help the plant.

While it's not generally recommended to grow them indoors, you can dig horseradish roots in the fall before the ground freezes and replant in a soil-filled container. Cover the roots with a 6- to 8-inch layer of damp sand or sawdust. Keep the soil evenly moist. Give the roots a cool, dark rest in cool temperatures from 50-60 degrees F./10-15 C. (such as in a garage). As the plants grow and leaves reach 8 inches in height, remove soil from the upper end of the main root and trim away small side roots. Then mound the soil up to its original level, repeating in a few weeks.

Here is an article for growing horseradish plants that you may find helpful: https://www.gardeningknowhow.com/edible/herbs/horseradish/growing-horseradish-how-to-grow-horseradish.htm

They will develop long roots like you have seen in the pictures, but it may take them 1-2 years to do so. In my own garden, I often lop the tops off the long roots that I harvest and replant them in my horseradish patch. They grow just fine; they just need a little more time to develop the kind of roots that you can harvest. If you are seeing growth from the leaves, this means the plant is happy and growing. Eventually it will produce the roots you are looking for.

There is no harm in leaving the flowers on the plant, they in fact will attract pollinators to the garden.

https://www.gardeningknowhow.com/edible/herbs/horseradish/growing-horseradish-how-to-grow-horseradish.htm