I’ve installed the Android Developer tools as per the official instructions.

Why doesn’t the tab appear in Windows->Preferences, as they said it should?

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This won’t fix your issue directly, but I suggest your take a look at the MOTODEV STUDIO for Android. Essentially, it’s a customized Eclipse IDE with all the Android tools preinstalled and many more of their own tools for app validation. I find their tools usefull, especially the code snippets and the app validator. I believe you will need to register to download, but it’s a quick and free process and you can always use a temporary email service. Anyway, this should allow you to get up and running until you find your bug (even if I don’t see why you’d switch back to plain eclipse).
MOTODEV STUDIO for ANDROID