Lets go step-by-step
1) I would recommend moving the code snippets that you have in the embed code blocks from the "Designer" area to Custom code area in the site settings. It will clean the site structure and you will not need to add that embed widgets to every page.
Here I will suggest using a Symbols. (Video Tutorial about Symbols: https://help.webflow.com/lesson/symbols)
All you will need is create Symbol from the menu, which is adjusted the best and then you can add this Symbol to any page. The best thing in using Symbols is that you are changing/fixing it on one page and all changes will be replicated to all other pages, where that Symbol is.
That blue "bar" is actually the bottom border, which appears on hover, set to Class "Menu link". So all elements, which will have this class name with have this blue line on hover. For now, you can't see this line on other links, because the Hero section has higher z-index than Navbar. If you will set Navbar position: relative, it will give Navbar higher z-index and blue lines will become visible.
Yet, I noticed, that you set menu link, which has a Current state to display: none. Not sure if it was done on purpose or by mistake.
Before moving to the fixing other problems you will need to remove class "Nav menu mobile" from the NAVBAR. Because this class has been used for the element inside this navbar and any further adjustments will make even bigger mess.
Let me know when you will apply this changes and we will move to the next questions.