For the question about mobility service dogs, this website gives a good overview.
http://www.servicedogcentral.org/content/node/507
You may not find much specific googling "mobility dog". They are basically service dogs that have been trained in tasks that will help someone who needs assistance with mobility. That could include acting as a steadying force or bracing in a specific way to help a person get up from a chair or the floor. Exactly what tasks are trained would depend on what the person
needs.
For the OP, we have not taken DD's SD to WDW, but we are at OKW frequently because it is our DVC home resort. Most of the time, the buses leaving OKW are not full, so boarding would not be a problem. All the stops have a wheelchair box painted on the pavement at the front of the bus stop, close to where the rear door would be.
You could stand in that box ( that is the direction for guests traveling with wheelchairs or
ECVs).
Most of the buses are the lowered floor buses and you would be fine going in the front door on those. The OKW bus stops have curbs and it sometimes takes the driver a bit to line up so that the ramp can be lowered in the curb cut area. For that reason, you may find it easier to just use the front door or to use the back door without the driver putting the ramp down.
The older buses have a steep set of stairs ( if I remember right, it's about 3 steps) whether getting on at the front or the back. I don't think our dog would have a problem with those steps because he is used to going up and down steps every day at our house.
I'm kind of wary about the lifts because I have gotten my foot scraped by the back of the lift while loading with my DD in her wheelchair. There are quite a few moving parts on the lifts, so I personally would either have the dog go up and down the steps or wait for another bus.
Leaving the parks with a wheelchair to go back to OKW, we don't usually run into overfilled buses. We usually try not to leave at peak times. After parades, at park closing or whatever, we usually just take our time leaving the park. That allows us to walk out with a small number of people, rather than in a large crowd. It also means that by the time we get to the bus stop, the first bus load(s) of people have left and we are often the first people at the stop.
Some of the park bus stops have a closed line, where guests enter the line and the only 'exit' is when the line gets to the front door of the bus. Epcot bus stops are like that. In those type of stops, you can tell the driver if you need to use the lift/ramp once you get to the front door.
If you get to the front door and see there will not be room to load your dog, you can step to the side and let others pass you so that you will be first to load on the next bus.
At other stops (and even at Epcot), you can wait about where the back door would be when the bus drives up so that the driver can see you.