HTML is used for structuring the content of a webpage, CSS is used for styling the appearance of the content, and JavaScript is used for making the content interactive and dynamic. They work together to create the complete user experience on a website.

Optimizing website performance involves several strategies including minifying CSS, JavaScript, and HTML files, compressing images, leveraging browser caching, using a Content Delivery Network (CDN), and optimizing the server response time. Analyzing and reducing the size of files and ensuring efficient load times are key practices.

Responsive design is an approach where a website's layout and content adjust fluidly to fit different screen sizes and resolutions, providing an optimal viewing experience across a range of devices, from desktops to smartphones. This is achieved using CSS media queries, flexible grid layouts, and responsive images.

The DOM is a programming interface for web documents which represents the structure of a document as a tree of objects. It allows developers to manipulate the structure, style, and content of websites dynamically using JavaScript. Browsers maintain a live DOM, so any changes made to the DOM are immediately reflected on the webpage.

Handling cross-browser compatibility involves testing the website on multiple browsers and using tools like Autoprefixer for CSS to ensure proper styling. Using feature detection libraries like Modernizr, adhering to web standards, writing clean and semantic HTML, and using polyfills for unsupported features also help in addressing compatibility issues.

Using frameworks like React or Angular offers several advantages including component-based architecture, better organization of code, improved efficiency, and faster development. They provide built-in tools and libraries for common tasks, facilitate easier state management, and enhance performance with features like virtual DOM (in React). These frameworks also have strong community support and extensive documentation.