[DEV-007] Add TypeScript check to required checks

[chev0004]

Outline

Summary

Adds a TypeScript no-emit check to the standard check pipeline so compile errors fail check:all and the pre-push hook.

---

Changes & Enhancements

• Type of Change:
  • Feature
  • Bugfix
  • Refactor
  • Chore
• Changes:
  • (tooling):
    • Add a typecheck script that runs tsc --noEmit.
    • Append typecheck to check:all, so the existing pre-push hook now covers TypeScript alongside Biome and i18n.

---

Page / App Changes

Affected Next.js Routes (Local):

Before (Screenshot)

After (Screenshot)

---

Component / Storybook Changes

Affected Storybook Components (Local):

Before (Screenshot)

After (Screenshot)

---

Not Doing

---

Other (Remarks)

---

Checklists

• Code is well-documented (comments, JSDoc, etc.).
• Existing comments were updated as needed.
• No out-of-scope changes are included.
• Write TODO comments where future work is required.
• Removed unnecessary debug code (e.g., console.log, debugger).
• Self-reviewed and tested locally.