Apply function style convention across API and web

[sam821203]

Description

Introduce and apply a consistent function style across the codebase: use arrow functions for all non-component code (helpers, API functions, handlers, mappers, hooks, async functions), and function declarations for React components and anything that returns JSX. A Cursor rule (.cursor/rules/function-style.mdc) documents the convention, and ESLint is updated to enforce it.

Proposed solution

• Use const name = (params): ReturnType => { ... } (or export const name = ...) for non-component code; use export const name = async (...) => ... for async.
• Use function ComponentName(props) for React components, subcomponents, and providers (e.g. pages, modals, DataBoundary, NotificationProvider).
• Add .cursor/rules/function-style.mdc describing the convention.
• Extend eslint.config.js to enforce the chosen styles.
• Refactor apps/api and apps/web to follow the convention (controllers, services, repositories, DTOs, API client, hooks, components, mocks).

Acceptance criteria

• Function style rule is documented in .cursor/rules/function-style.mdc.
• ESLint config enforces arrow functions for non-components and function declarations for React components.
• All API code (NestJS controllers, services, repositories, guards, interceptors, filters, config, mappers, DTOs) uses the arrow-function style where applicable.
• All web code (API client, error handler, hooks, helpers, mocks) uses arrow functions; React components use function declarations.
• Explicit return types and type annotations are preserved where they existed.
• Existing behavior is unchanged; refactor is style-only.

Additional context

Branch: refactor/function-style