As far as I can see, you have for example, module published in sidebar-a module positin but only on "Selected pages". If you would have template copy and assign it also on those same pages, you would see that layout config would be saved correctly. Try for example to publish some modules to all pages in those positions that you want to save and see if it will make a difference
Please try remove template from the site and then install it from the beginning. Some things are completely missing from template configuration so it seems that template didn't install correctly and completely