fix: repair failing tests and type errors across api and shared packages

[stooit]

Summary

Fixes all failing tests and type errors in the repository. Final state: 22/22 tests pass, tsc --noEmit clean (0 errors).

Bugs fixed

• Auth middleware case-sensitivity (packages/api/src/middleware/auth.ts): the public-method allow-list used lowercase "post", so POST requests (always uppercase per RFC 7231) were never matched and incorrectly required a token. Changed to "POST" and added .toUpperCase() normalisation.
• Missing import (packages/api/src/routes/users.ts): badRequest was called but never imported, throwing a ReferenceError and returning 500 instead of 400 on invalid input. Added it to the existing import from ../lib/errors.
• Inconsistent shared type (packages/shared/src/types.ts): User.userName did not match the username field used by the route handlers and tests. Renamed the type field to username.
• Unimplemented pagination utility (packages/shared/src/utils/pagination.ts): implemented paginate<T>() to satisfy the full test contract — correct page slice, second/partial pages, total/totalPages, page/pageSize, out-of-range returns empty data, and empty-array handling.
• Ambient types (tsconfig.json): registered the already-installed bun-types so process and bun:test resolve under tsc.

Verification

• bun test → 22 pass, 0 fail
• bunx tsc --noEmit → exit 0, no errors

Assumptions & notes

• The username/userName mismatch was resolved by changing the shared type (not the tests), since tests and route code both use username and test files must not be modified.
• No new dependencies were added; bun-types was already present in node_modules.
• No test files were modified.
• Changes are minimal and confined to the four flagged source files plus the tsconfig types entry.