Hello,
I have worked on various e-commerce software such as wordpress' woocommerce, Magento community edition, Magento enterprise edition, prestashop, x-cart, opencart etc.
And checked with your screen shots also. To fix this issue, just we need to make the changes in CSS. So that it will get aligned properly as you expected and also got removed the search feature as you expected.
It's pretty straight forward!
Warm Regards,
Subbu.