Is there a way to get the menu icon next to the logo?

Hi there,

The Site Logo in Customizer > Site Identity are loaded at original image size. You can either upload a smaller image or use the Logo size slider in the customizer to reduce it. You can then set the Customizer > Layout > Primary Navigation --> Location: float right.

This isn't putting the menu beside the Logo. It's still below it but to the right and the bar is not wide.

So you can use just the navigation as the Header:

https://docs.generatepress.com/article/navigation-as-a-header/

In the Customizer > Layout > Primary Navigation you can set Navigation Width and Inner Navigation Width to Full.