I always assumed the idea with a HA rush was the same as for cavalry, that you'd use espionage to take down cultural defense if at all so they don't get slowed down. But then i've never been much good at any kind of early rush above emperor.
Axe rush window on Emperor is pretty low. Unless im going BW due to forest and a copper pop in one of my cities, I'll just skip rushing.
Horse Archers on the other hand are pretty useful until AI get Longbow, thats a reasonable window for a rush.
I never ever rushed on Immortal, if I get boxed in I reload

