Forget about free domains if you want to do something remotely serious with them. A real domain is few dollars/year and AFAIK you could use it coupled with GitHub Pages for simple, static websites or with altervista.org if you need WordPress.
I second GitHub Pages!
I host a professional website/online CV there with a dot com domain for less than 10€/year.
There are Jekyll templates to do that or pay someone to make you some custom CSS and Html that's easy to maintain.
I feel like there's an inherent misunderstanding in your query. If you want to run on your own domain - you run your own stuff (for the most part). You need a webserver, that's about it.