GSOC-E2E-4: Add login test

[Geethegreat]

Issue:

Fixes #

Demo:

e2e-flow-4.mp4

Changes:

Changes

• Added a Playwright E2E test covering the user login flow.
• Creates a unique test user during test setup using the application's signup endpoint.
• Verifies that an existing user can successfully authenticate through the login UI.
• Validates that users are redirected away from the login page after successful authentication.
• Verifies authenticated navigation state by confirming the login link is no longer visible and the logged-in username is displayed.

I have verified that this pull request:

• has no linting errors (npm run lint)
• has no test errors (npm run test)
• has no typecheck errors (npm run typecheck)
• is from a uniquely-named feature branch and is up to date with the develop branch.
• is descriptively named and links to an issue number, i.e. Fixes #123
• meets the standards outlined in the accessibility guidelines