Stopping it expanding the header height when opened

Full width solves it (if you don’t mind it being full width)

Content >tab > Full Width = On

CSS to keep it just below the button

Advanced tab > Custom CSS

/*Dropdown without increasing header height*/
.elementor-nav-menu--dropdown.elementor-nav-menu__container {
	width: 200px;
	right: 0;
	position: absolute;
	top: 20px;    /*Adjust for where you want the top of the menu*/
	z-index: 2;
}

Pop up menu hidden behind a section it should be over

Select the section covering it and set its z-index to 1