refactor(server): migrate from try/catch to neverthrow Result types

[naaa760]

Summary

fix: #581

• this PR refactors the entire server codebase to replace all try/catch blocks with neverthrow's Result and ResultAsync types for functional, type-safe error handling. This standardizes error handling across all layers (authentication, database, business logic, providers) and ensures all expected failure paths are explicitly typed.

• The server previously mixed try/catch blocks and ad-hoc error handling across request, auth, and accounting layers, making control flow unpredictable and complicating typed error propagation. This refactoring:

• Standardizes error handling on a single, type-safe pattern

• Improves type safety by making all error paths explicit in function signatures

• Enhances maintainability with consistent error handling patterns

• Enables better error tracking with structured error types and context

Changes

New Error Infrastructure

• src/errors/types.ts: Defines standardized BaseError interface and custom error classes
• src/errors/result-helpers.ts: Utility functions for working with Result types
• src/errors/index.ts: Central export for all error types and helpers

Refactored Services

• Authentication layer now returns AppResultAsync
• Core services (EchoControlService, DbService, ModelRequestService) use Results
• Handlers check Result.isErr() instead of catching exceptions
• Providers removed try/catch blocks (errors handled upstream)
• Server error middleware handles BaseError, legacy HttpError, and generic Error types

Testing

• TypeScript compilation passes
• Build succeeds
• All existing functionality preserved
• Error handling paths verified

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