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

[stooit]

Summary

Fixes all failing tests and TypeScript errors in the multi-package API repo. Final state: 22 tests pass, 0 fail; tsc --noEmit clean.

Changes

• shared/types: Renamed User.userName → username (canonical name the tests assert against).
• shared/pagination: Implemented the paginate stub — 1-indexed pagination, start=(page-1)*size, returns data, page, pageSize, total, totalPages (Math.ceil(total/size), 0 on empty input; out-of-range pages return empty data).
• api/routes/users: Added missing badRequest to the ../lib/errors import (was causing a runtime ReferenceError on the 400 path).
• api/middleware/auth: Fixed case-sensitivity bug — public method list had "post" but Hono delivers uppercase method names, so POST /users was wrongly treated as protected.
• tsconfig (root): Added "types": ["bun-types"] to resolve bun:test and process type errors. bun-types was already present in node_modules — no new dependencies added.
• Removed three stale BUG:/TODO: comments that misdescribed the now-corrected code.

Verification

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

Assumptions

• Canonical field name is username (lowercase), since test files are the source of truth and may not be modified.
• bun-types in node_modules is acceptable to reference via tsconfig (no install performed, honouring the no-new-dependencies constraint).

Constraints honoured

No test files modified. No dependencies added. Only the changes required by the failing tests.