I want to write some Javascript classes which extend DOM nodes (so that I can then insert instances of my class directly into the DOM), but am having difficulty finding out
In 2020, you can easily create a custom element by extending HTML elements.
class AppDrawer extends HTMLElement {...}
window.customElements.define('app-drawer', AppDrawer);
// Or use an anonymous class if you don't want a named constructor in current scope.
window.customElements.define('app-drawer', class extends HTMLElement {...});
More info and reference: Custom Elements