The initial Gallery width is 1024px
I'm not sure what you mean by 'initial Gallery width'. If you set the gallery width to 1024px, then the gallery's width will always be 1024px. If you want the gallery's width to be responsive and fill the width of its parent container (even if its size changes, for example when the browser window is resized), then set the width to be 100% instead.
Does the text "Lite" in the right panel of Lightroom have any meaning, even if Juicebox Pro is chosen and is it reponsible for that.
The configuration options in the 'Lite' control panel correspond to the configuration options that are available to both Juicebox-Lite (the free version) and Juicebox-Pro. The 'Lite' options are listed here.
I installed the free Version and then the Pro.
That's fine. The Lite and Pro versions of the plugin can be installed and run side by side in the same version of Lightroom.
Also I see no possibility to generate multiple Sizes of the pictures
Unfortunately, the Lightroom plugin does not export multiple image sizes at once. The ability to generate a multi-size image gallery is a feature of JuiceboxBuilder-Pro only (although you could create a multi-size image gallery manually).
Is there now a possibility for not clipping the Thumbs?
All thumbnails in a gallery are the same size and if the aspect ratio of the source images does not match the thumbWidth x thumbHeight aspect ratio, cropping will occur. This cannot be changed.