One possible layout

On desktop, opt for a header with 2 or 3 rows. Due to less restrictive space requirements, you should surround major icons like the search box and shopping cart with a whitespace so they stand out more prominently.

There are various possible layouts, but below is a draft mockup of one of them:

Row 1

Contact info
Free shipping mention (it could be moved to the left)
“Help” link
Country selection

Row 2

Logo (clickable)
Search box (prominent, especially on the homepage)
Account icon
Cart icon

Row 3

Mega menu with the main categories as the level 1

A variation on the layout above

Please ignore the colors though, it’s only a low-fidelity wireframe.

separated top bar for the free shipping and help link
tagline
no country
labels on the icons
added the opening hours
moved the phone number
highlighted Specials menu item

How tall should the header be?

I looked at 12 well-known brands, and the average is 21.5% (measured with a browser ‘real estate’ of 1247px tall). I’d recommend keeping it below 20%.

Hyperlink the elements in the promo bar

What I like in the promo/announcement bar in the image below (Cotton Mills), above the menu bar, is that the 3 unique selling ‘bullets’ are clickable. It allows shoppers to quickly get more information by clicking directly on those.