What is the modern design and development approach towards a web application?

How do companies develop their web applications nowadays?

• Do they still develop use-cases, model, and class diagrams? What are these a part of (to know what term to search for)?
• Do they still use a design pattern? Where would that fall into the equation?
• Are the above necessary for a startup that's hoping to grow into a much bigger organization with teams of software engineers?
• Where do Waterfall and Agile fall into the equation?
• When are any of these unnecessary?

If you work in a company that does this, would you mind walking me through a summary of the workflow?

Any reading material provided as well would be much appreciated. Also would love some good sites that ELI5, if possible, and help me know what are the best practices. Finally, anything to help me choose the right design pattern and development methodology would also help, too. Thank you!