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

[stooit]

Summary

Fixes all failing tests and eliminates all tsc --noEmit type errors. bun test now reports 22 pass / 0 fail and tsc --noEmit exits clean.

Root causes fixed

• Pagination utility (shared): paginate<T>() in packages/shared/src/utils/pagination.ts was a stub that threw not implemented. Implemented per the test contract — correct page slice, second/partial-last page, total/totalPages/page/pageSize fields, empty-array and out-of-range-page handling.
• Auth middleware case bug: packages/api/src/middleware/auth.ts compared the HTTP method against lowercase "post", so POST /users was not treated as public. Fixed to uppercase "POST" to match c.req.method.
• Shared type field mismatch: the User type used userName while routes and tests used username. Aligned the shared type to the canonical username.
• Missing import: packages/api/src/routes/users.ts referenced badRequest without importing it. Added the import.
• Type-check config: added "types": ["bun-types"] to tsconfig.json so bun:test imports and Bun/Node globals (e.g. process) resolve under tsc --noEmit.

Verification

• bun test -> 22 pass, 0 fail
• tsc --noEmit -> clean (exit 0)
• Independent review pass confirmed correctness, no regressions.

Assumptions

• username (lowercase) is the canonical field name — determined by reading the unmodified test files.
• No test files modified; no new dependencies added (bun-types was already present).