I have personally never done vibration stuff, but I found in juce::PushNotifications::Notification there is some vibration functionality.
All this sounds totally possible to me. If the functionality you are looking for is not found in JUCE, I am sure you can easily access the Android API (via NDK) yourself in your JUCE app to achieve these things.
Here’s a forum thread on making your app wait: Best way to wait/sleep/delay?
Not sure exactly how to “background” a process, but you could always hide the application juce::Process::hide() and then when you want to notify the user you could juce::Process::makeForegroundProcess() and send the notification.
Hope this helps!