Thanks for the Details
Unfortunately I can not debug Custom Code in your Child theme.
The code changes made in the Starter Theme are more then a few lines, and I would need to consult a DEV member to debug what exactly is not updated in your Custom Child theme.
We can not do this, as we usually do not support Custom Coding.
Regarding your questions,
1. What does it mean? I have latest version of Stater Theme you suggested me to update.
Yes, but you have a Child theme and most probably you have customized some files where changes have been made by the Update.
If you do not change the Child theme Files, according the Updates, the parent theme has no possibility to apply the fix, as the Child theme Files have "priority"
This first answer to this post is very helpful to understand how, and when, you need to check your Child theme's files against Parent Theme Files after a released update:https://wordpress.org/support/topic/about-child-themes-parent-themes-and-updates
This is why I ask, if it is working with the original, parent theme enabled, then the issue must rely within the Child Theme Files.
2. If you know what must be updated then please tell me or i can give you credentials and you can see what's wrong. I can't wait every 2 days for replies. I have no idea what field in menu must be updated/changed.
As mentioned, I can not debug Custom Code, this is beyond the scope of our support.
At this point I would suggest you consider contacting one of our certified partners from this link:
https://toolset.com/consultant/
You will get the custom assistance you need to get on with your project.
I suggest to check what files you have changed in your Theme (Child)
Then compare those against the Parent Them's files
This way you will recognize where eventual changes in your Child theme are required.
3. How to know if plugin builds conflict with submenu?
I mentioned, you can check this by disabling the 3rd Party Plugins.
If the issue goes away by doing so, you will know that one of the plugins builds issues.
Then you can find it by re-enabling one by one and find when the issue comes back.
I apologize that I can not debug the Custom Code.
Please let me know if you need further infos about this.
Thank you for your patience.