How do you export table data without running the query in the Editor?
Use Database | Export | Export Utility Wizard. This will create a dmp file that recreates the table and inserts the data. Note that a license with DB Admin module is required for this feature.
Go into Schema browser, select a table and right click. Choose either Export Data or Save As. Set options and run. Neither of these will include table create scripts.
Export Data will create a script with insert statements for your data.
Save As will allow you to choose what format you want to export your data as.
These instructions are based off of Toad 9.7. Similar features exist in older versions of Toad also, under the Database | Export menu and in the Schema Browser.