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WebOS is based entirely on Palm's internal code. I do not believe any part of PalmOS or Access Linux Platform is part of it. The "Classic" part of WebOS is an application licensed from someone else who apparently has a license from Access to write said emulator.
However, Classic is not part of WebOS, at least not on my Pre .
The BeOS source code is still owned by Access, as is Cobalt and the other PalmOS iterations. One of their representatives has posted here multiple times, and the likelihood of them open-sourcing BeOS or the other code they possess far outweighs the monetary benefit they would derive from it (not much).