To use EMH, do you have to be at the park before they start? I was thinking it would be nice to hop over to Epcot for the EMH (since its the latest) the night it is offered, but wasn't planning on being at that park that day. Can we just show up once they start?

Yes, you can show up after official park closing (or in other words, after the start of EMH). I don't know if there is a point when they stop allowing admission into the parks before closing time...

Just be sure that you have "hoppers" as part of your ticket package-if you've been in another of the 3 parks that day. I know that seems fairly obvious but just thought I'd mention it.