*At what age do they begin to feel the "Disney Magic?"
When they can talk and tell you about what happened more than 1 day ago.
* At what age are they able to enjoy the rides?
When they can tell you their version of what they are seeing. Do they know the story of Snow White, Peter Pan, Little Mermaid, Lion King... If they have no idea what it's all about they'd be just as happy playing in their own back yard.
* At what age are the parents able to relax and enjoy their WDW vacation?
I think it depends on the kid(s). If your child is high maintenance going to Disney could be a lot of work for you, it might pay to wait for them to grow up a little. On the other hand if you have easygoing kids, like I do,

it will come much sooner.
Personally, I wouldn't bring a kid to WDW until they are out of diapers and able to walk long distances and/or stand in line for a long time without distress of some kind. If they are at a stage where all they will eat is Kraft Mac & Cheese and bananas Id think twice. It breaks my heart to see frazzled red-faced parents struggle with screaming unhappy kids as they drag them through the parks. I can almost read their mind, I dont see whats so GREAT about this! Get me outta here!
Remember though that Disney isnt just for kids, you need to have a vacation too. If you decide ahead of time that if someone in your group isnt having fun youll go back to your resort and hang out by the pool or take a nap it can still be a relaxing vacation even with very small children. Our family routinely splits up so folks can do their own thing.