I have just installed GeneratePress theme and premium extensions. I also have layouts set up by the Toolset Starter theme demo installer (but I removed Toolset theme). I try to configure theme options for a layout for pages with GeneratePress theme being active, but they are not saved. I mean I see a green mark telling that the changes were saved but when I go back to layout theme options again I don't see these changes, instead just default setting again.
When I do changes to the theme options within a particular page (not on a layout), or even within the editor (every time I make a single-letter change for example and update the page), the default layout is being removed from a page and the side Template Layout panel says "Don't use a layout". This is the same situation I already described but for products: https://toolset.com/forums/topic/editing-product-reset-layout-assignment/, but now I moved with a site to online environment, with fresh WordPress installation.
Concerning the second issue, I recorded a short video where I update a page by typing some text. After update, the layout linkage is broken: hidden link
That's certainly not expected, and you say this is a new installation, and from your debug info I don't see many other plugins that might be responsible.
I installed the copy of your site locally and couldn't reproduce the problems.
I was able to change the theme options for a Layout and they persisted, and I was able to edit a page with a Layout assigned and that assignment persisted upon saving (see screenshots).
While updating these pages on your site, can you check the browser console for any JS errors.
Also enable debugging to test for PHP warnings or errors.
If you haven't already, turn on the debug log by editing your wp-config.php file and change the line with WP_DEBUG like so:
That will create a debug.log file in your wp-content directory which you can examine in any text editor. Try visiting the same pages where the problem occurs and then inspect the log. If you don't find the debug.log file it means it didn't generate any warnings or errors.
I did that, but no debug file was created... Anyway I know the reason now: I use MS Edge and in that browser somehow I encounter the above issues. I checked with Firefox and it worked without problems. Nevertheless, it'd be worth fixing so that no such issues are present for Edge.