Chat: Typing Indicators

Provide real-time feedback when a user is typing a message.

Task: Emit typing events via Socket.io from the client and broadcast to other participants.

Start at the chat client and the existing Socket.io event flow. Trace how messages reach other participants, then add the typing event path so participants receive real-time typing feedback. Done means a client emits typing events and other participants receive the broadcast while a user types.

Written by the indexing model from the issue text.