My questions are:
1. Is this word still in use?
2. Is it a regional (UK?) thing?
3. Do sunk fences still exist? I can't recall having seen anything, anywhere, that would qualify for the description.
________________________________________________

1. Absolutely!
2. Quite conceivably a UK thing. I seem to recollect that they were a particular favourite of Capability Brown.
3. They definitely still exist, and can be found at any number of stately homes in the UK and also in some larger private houses. As Bingley says, the basic aim was to create a formal garden that the livestock couldn't get into, without in anyway compromising the view. I think it's usually a brick wall, rather than a fence, creating a sheer drop that the animals can't climb. On the other side, the land slopes gradually to the base of the wall - prevents the animals from being able to jump from one side of the ditch to the other.