Can you shooting and downloading photos simultaneously?

No you cannot. You have to switch modes in order to shoot photos or download photos. They are in different camera mode. In order to switch mode, you can refer to the code below.

mCamera.setMode(SettingsDefinitions.CameraMode.SHOOT_PHOTO, new CommonCallbacks.CompletionCallback() {
    @Override
    public void onResult(DJIError djiError) {
        if (djiError != null) {
            showToast("can't change mode of camera, error:"+djiError.getDescription());
        }
    }
})

mCamera.setMode(SettingsDefinitions.CameraMode.MEDIA_DOWNLOAD, new CommonCallbacks.CompletionCallback() {
    @Override
    public void onResult(DJIError djiError) {
        if(djiError != null){
            showToast("can't change mode of camera, error:"+djiError.getDescription());
    }
});