1) appMobi should always run in fullscreen, do you mean removing the status bar? If so the 'hide status bar' action in the appMobi object should do the job.
2) Re: tabs - this has been requested a few times, but is difficult to change because of the UI library we use. I'll try look in to it soon.
3) Re: 'Free form tiled backgrounds': can't you usually fake these effects with a cleverly designed rectangle texture? The reason the tiled background is rectangle only is because the underlying graphics hardware is very very fast at rendering that - other shapes would be a lot slower to render, so it's best to use rectangles in the Tiled Background object.
4) For issues with CocoonJS, it is best to contact Ludei. Only they can fix problems in the CocoonJS launcher.
Hope that helps - sorry for the delay.