Menu saves only half and is not displayed in the top menu

This theme allows only 3 levels of sub-menu, please see the instructions if you would like to add more levels of menu depth:

https://mtomas.com/389/tiny-forge-framework-child-themes-comprehensive-guide#Increase-depth-level-for-sub-items-of-top-navigation-menu

I am using only 2 levels. (primary and then sub and sub-sub)
Is there also a maximum to the number of ‘items’ in each level?

Not that I know. If menu would be very long, most likely it would not fit on the screen. If you cannot see other levels, it is possible that some of the CSS changes that you did might caused unexpected behavior.

The easiest way to test this – try upload fresh Parent theme (or parent+fresh child) and see if the problem persist.