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Yes, you can leave it. Hopefully, you will also be taking care to keep the plant under control as well. Once the plant starts growing well, it can quickly become invasive. This article will help with its care: https://www.gardeningknowhow.com/edible/herbs/horseradish/growing-horseradish-how-to-grow-horseradish.htm

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. You can trim away any flowers that may appear though.

There are hundreds and hundreds of Horseradish recipes available!
Inquire at your local food coops or Farmer's Markets for gardeners who make their own sauces.
They may be willing to mentor you.
Here are some links with more information.

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

You do not need to break off the sideshoots to make the plant larger. You can divide the plant and replant the sideshoots to make more plants and get a larger harvest.

Brown leaves are usually symptoms of under- or overwatering, and it sounds like the recent bout of rain that you've had could be the culprit here -- but there are other possibilities. This article outlines the common causes for leaf browning in vegetable plants and some remedies:

https://www.gardeningknowhow.com/edible/vegetables/vgen/brown-leaves-on-vegetables.htm

It is not necessary to trim back horseradish leaves. For neatness, you can trim back any leaves that have turned completely brown. Leave any green leaves on the plant so it can continue storing energy as late as possible.