I'm not sure of the exact time of the last RER but I think it's something like 00:15 so if you were staying until New Year's Eve it would be a bit of a rush (unless they run the trains later on this night?

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How about seeing if you can book one night at one of the offsite hotels; Explorers, Magic Circus, Dream Castle or Kyriad ~ the latter is the cheapest. Although they're a bit farther than the Disney hotels and won't be as cheap as the ones a train ride away they are serviced by free shuttle buses which will run until 1am on New Year's Eve (and if you miss the bus it's only a 25 minute walk).
As for being there on New Year's Eve, it does get very, very busy but I understand the atmosphere is fantastic. A few DISers have been there at New Year so have a look at some of the trip reports.
You have three options to get to the airport ~ the VEA bus, TGV fast train or book a private shuttle. Unless you book the private shuttle, it will be easier to get to the airport from Disney. There's lots of info on all in The Very Useful Info sticky but just ask away if you want to know more.
