The info you can export out of QB Online is limited at this time. If there's important info which you're finding hard to export out of QBO, don't hesitate to reach out for help. Once you've exported you QBO file as CSV files, you can import them to QB Desktop.
To import a list in the CSV format, you need to install the QuickBooks Import Excel and CSV toolkit. The toolkit contains:
Import from Excel and CSV Manual – a PDF document with instructions, best practices, and notes.
CSV Examples Folder – 4 sample CSV files, one for each type of list.
XLS Example Folder – a sample Excel file containing 4 worksheets, one for each type of list.
Allowed fields – a reference guide explaining what fields are available for Excel and CSV import.
Steps to install the QuickBooks Import Excel and CSV toolkit