Hi,
Your custom jQuery were conflicting with native Kentico jQuery, take a look at:
How to register and use the native Kentico jQuery libraryUnfortunately, this customization of Tab Menu Web Part is possible only via the Source code. Let me know if you have the license with the source code and I will discuss it with development team.
On the other side, you can create a custom Web Part and take inspiration from the in-built Tab Menu Web Part.
Best regards,
Martin Danko