JetBrains: Update docs

Created by: vdavid

Updated a few docs pages, then the README for the JetBrains project, to be up-to-date with the latest changes, and have production-ready docs quality. I think it's good now.

Also fixed VS Code links once I was there.

Adding @philipp-spiess as a reviewer for the JetBrains part and @abeatrix for the tiny VS Code-related changes.

Test plan

Only docs changes, so the code review should do it.

App preview:

• Web
• Storybook

Check out the client app preview documentation to learn more.