test(api): refactor client tests to use simplified send_message API

- Remove convenience method tests (get_game_state, go_to_menu, etc.)
- Replace with direct send_message calls for all API functions
- Update test method names to reflect new API structure
- Add comprehensive test coverage for send_message with all API functions
- Update parameter passing to use structured arguments
- Fix socket configuration tests to handle class attributes
- Maintain backward compatibility testing for error handling

Author: S1M0N38