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Since this is a gardening forum, you would be able to find better information from an agricultural or farming source. However, as I grew up on a horse farm, and my dad grew lots of hay, I'll share my personal experience. Part of your decision of what to plant is based on your market, what kind of animal the hay is meant for, and what kind of hay people want to buy. Agricultural studies are not in total agreement as to the best hay for horses. I learned that 100% alfalfa was not good because it could cause digestive problems, and we always used a mix of timothy and alfalfa. Hope this is of some help to you, but I really recommend you do some research on the farming and university sites.

You should not turn them out into an alfalfa field, as they tend to overeat it that way and will bloat up and can die from it. They will at the least get severe diarrhea. The alfalfa will start to form little purple blooms and is ready to cut. Let it dry in the cut rows for a few days then have it baled or rake up and stack into a mounded stack. Then it can be fed to your cows with less problems and they will still get loads of nutritional benefit from it.

The term cool season perennial doesn't refer to the climate required for growing, but the temperature required for germination. These plants need cool temperatures and short day length to induce sprouting. They are usually planted in the fall in milder climates, and in the spring in colder climates. Alfalfa is quite hardy while maturing, withstanding high temperatures and drought. While it's not a grass (it is a legume,) this article can explain a bit more about cool season and warm season: https://www.gardeningknowhow.com/lawn-care/lgen/what-is-cool-grass.htm

Alfalfa plants adapt fairly easy to most climate and soil conditions but to answer your questions:

1. Cool-season simply means the plants typically grow better in cooler conditions during spring and fall, such as the northernmost states in the U.S. That is not to say, however, that alfalfa cannot be grown in other areas like the South.

2. As for the soil, alfalfa prefers soil that is deep and well draining with a pH of around 6.5 to 6.8.

Since Alfalfa is a cool cover crop you will want to plant in the fall in your region.

Here are a few links with more information.

https://www.gardeningknowhow.com/edible/vegetables/alfalfa/growing-alfalfa.htm

Club root disease affects only plants in the Brassicaceae family (cruciferous vegetables), so you can safely plant anything from other plant families, including alfalfa.
https://www.gardeningknowhow.com/edible/vegetables/vgen/cruciferous-vegetables.htm

Avoid growing any Brassicaceae in the infested soil, because the organism that causes the disease is though to survive in the soil for years (some say 10 to 20 years!). Here is more information about this disease:
http://vegetablemdonline.ppath.cornell.edu/factsheets/Crucifers_Clubroot.htm

According to the linked article, legumes like Alfalfa and a few other plants can also carry the disease, but it doesn't damage those plants' health.

Seeds can spread clubroot only if they are dirty when they arrive, but the most common methods of spread are by wind and by transport of soil.