Disney World is in the US, so all the US laws apply to Disney for everything, including service animal access. In other words, no certification/paperwork/vest/etc. is required (but I highly recommend a vest if you don't use a harness, as it keeps access problems and questions and drive-by pettings down). They cannot ask your what your disability is or require to see any paperwork or ID cards or such about it or the dog. (Never give these out in the US - it will teach them to look for them with every team and not every team has them or carries them with them even if they do have them.) They cannot charge you an extra fee (i.e. pet fee) - but if your dog wrecks something and a non-disabled customer would be charged for doing it, you will be charged.
And, of course, they can ask you the three questions the law allows: (1) Are you disabled? (2) Is this your service animal? (3) What tasks has he been trained to do that mitigate your disability?
There are some rides, such as Splash Mountain and Space Mountain and Thunder Mountain (all the -Mountains!), that service dogs are not allowed on due to safety reasons. This is legal because it is for safety's sake. On those rides, somebody else in your party will have to hold your service dog for you while you ride. When you finish the ride, you stay with your service dog while the other person/people ride (or vice versa). There are plenty of great rides your service dog can enjoy, though, like Pirates of the Caribbean and "it's a small world" (if that music doesn't get to them first, ahahaha!!!).
Like somebody else said, companion dogs are NOT service dogs. Service dogs have been highly trained to do tasks for a specific disabled person (their owner), such as dogs guiding the blind/visually impaired, dogs helping people with mobility issues to walk, dogs alerting and/or responding to the onset of medical episodes like seizures, etc. They are also highly trained in public access skills and are "invisible" when in public. If your dog is a companion dog or emotional support pet or therapy dog or other type of pet, he is not allowed to stay in Disney's hotels (except a few particular spots at the Fort Wilderness Campgrounds) or enter Disney's parks, but Disney does have kennels available for dogs to stay in (I think they might be only for during the day, not overnight, but somebody correct me if I'm wrong).
And, yes, if you are going on an airplane and want to sit in the bulkhead seat, the ACAA gives people with service animals (as well as people with legs/prosthetics that can't bend) priority over other passengers for the bulkhead seats (UNLESS they are exit rows, in which case no disabled person is allowed to sit there, plus the dog would be against the safety regulations in an exit row). Bulkheads may or may not give your dog extra floorspace, but they will prevent him from having to try being stuffed under a seat or getting stuck under one (especially if he's in a harness). If you need the bulkhead for your service animal, you must call the airline to request it 24 hours or more in advance of your flight (I normally call right after I make my flight reservation online).
The ACAA rules (for airplanes) differ slightly from the ADA rules in that if there is no visual evidence that your dog is your service dog and your verbal word is not credible, they CAN require you to produce documentation that your dog is, indeed, a service animal. However, they are not s'posed to ask for this info. if your dog is harnessed (or vested) and behaving like a service dog should. Like with anything else in America, they CANNOT ask your what your disability is (though some do try)!!! That is legally confidential information that only a JUDGE in a COURT can require!
If you have any other questions, just ask! There are a few of us here on the boards who have service dogs and have been to Disney with them.