How can I work on the same query if the connection is lost (timeout) and then I reconnect?

Depends what you are doing. If you ran a stored procedure than it will finish but that also depends how you coded transaction handling in the stored procedure. If you are just doing a simple select of an insert and the connection is broken. Then Sqlserver will rollback in case of an insert. A simple select will be terminated. If you want to guarantee the fact that you want to be able to work on after a connection loss then you should transfer your SQL into a stored procedure and in that stored procedure you program that the next time the stored procedure is executed ( after a connection loss) that it will continue the job it was doing by looking into a status table where it can pick up were it left.

To adjust the query timeout, access the server properties through the right-click menu, navigate to Remote server connections, and locate the remote query timeout field. Enter a large number in that field.

When this happens to me in SQL Server Management Studio, I just have to type USE {your database name here} and then select only that statement and execute it. This re-establishes my connection and I can continue to work on my query.

You can change the timeout for the query: In the object explorer, right click a server and select properties. Go to Remote server connections, look for the remote query timeout field; then type or choose a value from 0 to 2,147,483,647 to establish the maximum number secs that SQL Server will wait before timing out