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

[stooit]

Summary

Fixes all failing tests (now 22 pass / 0 fail) and eliminates all tsc --noEmit type errors across the api and shared packages.

Changes

• packages/shared/src/utils/pagination.ts — implemented the paginate() stub (page slicing, total, totalPages, page/pageSize), with guards so totalPages is 0 when total === 0 or size === 0 (no Infinity).
• packages/shared/src/types.ts — renamed User.userName → User.username to match the canonical field name the tests expect.
• packages/api/src/routes/users.ts — added the missing badRequest import and aligned field usage to username.
• packages/api/src/middleware/auth.ts — fixed a case-sensitivity bug in the public-methods list (post → POST) so POST /users is correctly treated as a public, no-token route.
• tsconfig.json — added "types": ["bun-types"] to resolve bun:test module and process global type errors (uses the already-installed bun-types; no new dependencies).

Verification

• bun test → 22 pass, 0 fail (4 files)
• bunx tsc --noEmit → clean, zero errors

Assumptions

• The canonical user field is username (all-lowercase), since the tests use it throughout and tests are authoritative / were not modified.
• No test files were modified and no dependencies were added, per task constraints.