I’m not sure where you’re lost exactly, but all you would need to do is add a widget to a widget area and it shows up on the homepage (here’s a guide – https://appfinite.com/topic/epik-widgets-guide/).
There are code snippets and tutorials at the top of this forum – https://appfinite.com/forum/epik-theme-support/ (Yellow Sticky Posts) which explains exactly how the demo was made. Use text widgets if you want to add custom code, text, etc.
As far as separating your custom code, you could either create a new style.css and functions file and link to them from the main style.css and functions file…..or you could just add your custom code at the bottom of either of those files to keep it separate and organized.
The child theme itself doesn’t add a header.png file, but WordPress/Genesis does. You should be able to remove it by clicking the Remove option in your header uploader then reupload your logo if you plan to use one.
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