We've stayed at the adagio twice, so I'll try & help!
As has been said there is a car park under the hotel, but we saw loads of people parking on the streets outside the hotel.
The adagio to the rer station is quite a walk, you can either walk right through the shopping centre (about 10-15 mins) or walk alon the streets around the shopping centre.
The shuttles can get packed, but it depends on the time of day, we found they werent that packed in the early morning as most people staying there aren't eligible for emh so don't go till getting on for 10am, we had dream annual passes so could get there for 8am. The biggest problem we had was the metal bars that they have in the door ways on the buses, we couldn't fit our double pushchair through some of them, so would have to wait for the next bus! It was easier to do that than unload our 1 & 2 yr old, all the stuff off the pushchair & try & keep track of the other 3 children!
The first time we stayed there we used to walk most days, it was lovely weather (we went in August twice) the way we went was come out of the adagio, turn left, walk a few metres then cross the road, walk through a little square, there were a few shops here, then carry on walking down the long road, they were building on the right hand side, at the very end of the long road there is a police headquarters? building, cross the road after that and turn
right, then you come to a roundabout (I think it was!) turn left at the roundabout and walk down that road, you will see the newport bay hotel to your right. Carry on down that road, you will pass an old car park on your left & the roads leads you to a multistorey car park right in between the studios & the Disney village. I would say the walk is about 25-30 mins.
As far as I remember, no where at val d'europe is open before 10 am.
Hope this helps & you have a lovely trip!