I've added pages to my hub site and added links to the second-tier header via custom html/css. However, no matter what page I'm currently on, the home navigation tab (<li>) is still set to active. Am I doing something wrong to link to these pages? Or how can I overwrite this css class?
Unfortunately, having an active page marked in a custom Header/CSS implementation is not possible. This is because pages inherit the site's branding to maintain consistency. You may have noticed that you don't have the ability to customize header/footer/theme at the page level and so if you're setting an active class on the homepage for a link, it will stay that way on all other pages because you set it from the homepage.
Are there any plans to make this possible? It greatly reduces the benefits of a custom header if you can't easily highlight to the user what page they are currently on.
As of now, there are not any plans on our side for adding WYSIWYG (What You See Is What You Get) controls to custom headers. We make custom headers an available option, but we do prefer you use our suggested layouts if you want us to manage mobile and active states for you. If you, the author, of a custom header can achieve a CSS only solution for active pages - then it's entirely possible. You might look into using a[href] as an attribute selector in your CSS.