What a great question as point of departure
Your derivation of
oak is OK in
this book:
Derry (doire) is an oak grove and this placename is used on its own or in combination with other words in many places in Ireland. For example Derrydrummuck townland in the Aghaderg Parish of County Down comes from Doire Droma Muc or 'the oakwood of the ridge of the pigs'.
Of course Derry and Down also exist as counties in Ulster. That site don't really get down with
Down as a name, though they're all over the map with place names in said county.
Now on to the dance... your get-down-and-derry derivations are fascinating, e. A good source for sleuth.
Quoth another source:
Etymologists have traced this phrase back to Norman England, to the Danish days, and even to the Saxon epoch, only to have it elude them at last. It is considered probable that the words are of Druidic origin.
(--Shakespeare in Music -Louis Charles Elson)
Elson herein appends a meaning of "burden" to
derry down.
I'm kinda more interested in the Hey or Hay.
This folkdance site notes three claims of varying dubidity, claiming a
hay (noun, as a dance figure) is "often known as a reel in Britain" (although this does not square with what I know from music).
More:
Spelled variously, hay, haye, and hey, this is a very old country dance, usually a round one, and Dr. Johnson, in his dictionary, suggests that it may have been so called because it was originally danced round a haycock. In Love's Labour's Lost, Dull, the constable says, "I will play on the tabor to the Worthies and let them dance the Hay."
-and-
Arbeau describes the hay exactly as it is still done in English country dancing today, as well as elsewhere in Europe. ... The term may be pictorially derived from "la haye," a French for an artificial hedge, "formed of upright wooden stakes interlaced with transverse strands consisting of thin supple stems"
Ah, there is nothing like playing on the tabor to the Worthies. Only in English.
But my knowledge of the
antic hey comes via Barbara Walker:
Dance step of the medieval Carnival king; antico from Latin antiquus... Carnival "antics" were connected with the Old Religions whose sacred processions were often accompanied by clowns deliberately making obscene gestures and jokes to heighten the spirit of revelry. The hey was and is a figure-eight pattern paced on the ground, the sign meaning 'infinity' in Hindu-Arabic numeral systems and their descendant, modern mathematics. (Womens' Encyclopedia of Myths and Secrets)
Apparently next time we hear someone say "hey" we must ask them to be more specific.
What of Hoedown?
One strange-looking site to which I've lost the URL claims:
Hoedown Style
This is American by derivation. Back in the days of the mid-west dirt farmers who toiled with their horse and plough in the 1800s - the farmers, at the end of the day, would literally put their hoe's (sic) down and throw a dance with whatever instruments they had to hand - hence the name Hoedown.
"Hey" and even "ho down" (the plot thickens) may be seen in
this song transcription, along with another interesting twist, the insertion of the similarly seemingly meaningless "to my". Usually rendered as "to me" on the offbeat just before the "one" beat of a chorus, "to me" gets used very commonly in the rhythmic work song of sea chanteys, where the lead singer signals that on the next beat the action (say, hoisting a sail) goes towards him.
You can see how disappointed I would be, see avatar. 
