What is the etymological connection between draught beer and draught horses? I know that draught horses used to pull carts full of kegs of draught beer - is that a coincidence or not?
The OED has the horse definition of “draught” appearing in English before the beer definition.
“Draught” originally referred to pulling a plough.
Ahh, silent letters.
It’s an older form relating to the verb “to draw” e.g. the horse draws the plough; you draw the beer from the keg.
Of course where I come from draft horses pull plows and we drink draft beer.
That’s my spelling too. Draft horses draw a dray, among other things. Both of the italics come the same root, the OE dragon - to draw or drag (from ME draggen from ON draga).
So is ‘draught’ incorrect, or merely old-fashioned?
I admit I was tempted to write ‘connexion’ above.
Horses are really bad spellers, particularly when they are drunk.
All that beer you’re drinking is clearly affecting your spelling ability. 
It’s the preferred spelling given in all of the dictionaries that I have consulted.
So would it be draughting or drafting (if one is drawing a buiding or suchlike)?
“Draft” appears to be the preferred spelling in the real world. Googling “draught horse” got 170,796 hits; “draft horse” got 1,256,401 hits.
The various draft horse associations and magazines, such as The Draft Horse Journal, in the United States tend to spell it “draft”. On the other hand, it’s spelled “draught” by all the Irish Draught Horse Societies around the world, including the North American association.
My guess would be that “draft” is the more common American spelling, while “draught” is still standard in the British Commonwealth.
That’s what my Merriam-Webster Collegiate says.
Those Daught Irish horses and their drunken spelling…