Upgrade Next.js to v16

[nabramow]

Bump next, eslint-config-next, and related packages from 15.5.x to 16.x.

This also unblocks removing the ESLint FlatCompat (see below).

Here's the migration guide: https://nextjs.org/docs/app/guides/upgrading/version-16

TODO:

• Upgrade next → ^16 and eslint-config-next → ^16.
• Replace the FlatCompat-based ESLint setup with Next 16's native flat configs, dropping the @eslint/eslintrc dep and the manual @typescript-eslint/parser block:

Note / risk
Next 16's flat config bundles react, react-hooks, import, and jsx-a11y with their recommended rulesets enabled, so new lint warnings/errors are expected — this PR includes the re-lint pass to resolve them.

To test:

• yarn build passes
• yarn lint passes (--max-warnings=0 in CI)
• yarn test passes
• Run app and click around, log in and out, check main functionalities, make sure nothing broke