Will adding a new column in Google sheets to the right of the integrated data from jotform? How about another sheet?

Please note that the Google Sheet integration may be broken if you edit the integrated sheet, the current integration  will add a new row if a new submission is received, but it's not possible to add new columns. 

I would recommend you to take a look on the Zapier integration between JotForm and Google Sheets: https://zapier.com/zapbook/google-sheets/jotform/ 

I hope this helps. 

Sending me to another site is not much help.  Does JotForm not support the integration with Google Sheets?

Yes, JotForm has indeed a native integration with Google Sheets, please read more about it on this guide: https://www.jotform.com/help/228-How-to-Integrate-Form-with-Google-Spreadsheet 

But as explained above, we do recommend to avoid making any change on the integrated sheet in order to keep it synced with new submissions. Once you integrate your form a brand new sheet will be created on your account and a new submission will end up in a new row on the sheet.