Hey @Sash!
I hope I can provide some clarity 
We recently rolled out a new unified menu experience to Studio, as announced here. Which is why you might be experiencing things that are slightly different to previous tutorials etc.
Here’s a few articles that give more info about working with the new menus:
To answer some of your questions…
You’ll need to work with some of the options in the panel to adjust the height, such as “Item padding”.
It’s important to remember that Studio and Wix Editor are designed for different needs. Studio is built with agencies, freelancers, and enterprise users in mind, offering more flexible and customizable options - whereas Wix Editor focuses on out-of-the-box functionality.
The example you shared isn’t using Stacks. There’s currently no relation between the Menu and “Contact” Button. When the menu expands, it doesn’t push, because there’s no relation to the element under it.
By creating a Stack, you tell the elements “I want you to always have this spacing between each other”, so when the Menu opens, the Stack grows in height, and the Contact button always remains underneath it.
I recommend taking a look at this video, which goes into more details about what Stacks are and how they work - https://www.youtube.com/watch?v=ESFNNso4Vlo
Headers don’t have the shadow option as they are Sections - as Sections are the base layer of the page, and reach side to side, for the most part, shadows are redundant (as Sections always touch the side of the screen and arranged tightly to one another). As suggested in that request, using a stretched container provides the same functionality
I guess you’re talking about the “Flyout” option, right? I’ve asked the team and will circle round once I have more info.
I appreciate all the insights shared in this topic and shared a link with the team Thanks for sharing!