Migrate lint script for Next.js 16

[tongshu2023]

Summary:

• Replace the removed Next.js lint command with the ESLint CLI.
• Move the project to ESLint flat config for the existing Next.js 16 ESLint config package.
• Upgrade ESLint to v9 to satisfy eslint-config-next 16 peer requirements.
• Replace internal home-page anchor links in Header and Footer with Next Link so the migrated lint command passes.

Why:
The project uses Next.js 16, where next lint has been removed. Running the existing lint script currently treats lint as a project directory and fails before ESLint can run. eslint-config-next 16 also declares ESLint 9 as its peer requirement, while the project still had ESLint 8.

Verification:

• .\node_modules\.bin\next.cmd lint fails with Invalid project directory provided, no such directory: ...\llamatutor\lint before this change.
• .\node_modules\.bin\eslint.cmd . passes after this change.
• git diff --check passed.

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