Well, these days I'm also lurking in the MA, CT and RI forums

We still want to migrate back to New England, but lately we're having second thoughts about moving to Vermont and dealing with 7 months of cold weather! We're looking for a happy medium. So far, South Kingstown, RI, right near the ocean is looking good to me

Summers in the 80s and Januarys in the 30s. Also not in Vermont's favor is their 8.5% state income tax, and $19.00/$1000 or higher property tax rates. We're not ruling it out but so far my Realtor.com Saved Properties list has 2 Vermont houses and about 16 RI, MA and CT houses
With respect to VA laws, if memory serves, you must sell $1,000 or more in livestock, produce or farm products during a calendar year. If you didn't but if you would have otherwise (perhaps nobody bought any of your sale horses, or due to poor weather conditions your crop was lost etc) then you still qualify. Virginia has a very inclusive definition of a farm and horse farms have special protections written into the Virginia code to protect equine operations. As long as you abide by the code (one example: having liability waivers signed by students) then you are in almost all cases protected from liability.
There is no specific process for registering a farm name other than the traditional business channels such as organizing a company (most often an LLC) and registering your trademark. Some farmers never form an LLC and just file the appropriate schedules on their personal returns (Schedule F?). If you file the appropriate schedules you aren't obligated to pay sales tax to some vendors (feed stores, tractor sales, etc). You also get to use those coveted FARM USE tags on virtually anything with wheels that you use exclusively for farm use. And no, BMWs don't count. I checked.
Life is much simpler without an LLC. However, the LLC does offer a layer of protection. Example: the horse business is an "LLC" but leases the land/facilities from you the owner. This would insulate your real estate and prevent it from being subject to a lawsuit; after all, you might be protected from liability from a lawsuit but the lawsuit and any appeals could take years and during that time, if you're not insulated, you would be prevented from selling the property. That's my understanding, anyway. Better to be safe than sorry.
Picking a name is fun

We settled on ours on a 10 hour drive to Vermont in early 2005. We brainstormed through a pile of corny ones at first and then it went downhill for awhile (Snob Hill Farm, Tax Shelter Acres, Money Pit Meadows). But since we're on a hill and one of the first things you see when you drive in is a very large white oak, we settled on "Oak Hill Stables". Not very imaginative but it has grown on me.
Sean