How to add checkboxes to a Pandas Dataframe table using Flask?
I would do it by workaround here.

Create html combo box in template.html
Populate combobox with values from database in routes.py
Also in routes.py read combobox selected value and according to that value change sql request to database - e.g. value 1 - sql query without where condition else add where condition name = (in) selected value(s)
When you send request you can in routes.py choose if you want render_template with different data or if you submit to another route which will export you data to text file (you can stay in pandas and use to_csv)

Also your question contain multiple problems. It would be better to be more specific.
If you would like only to add checkboxes as another column in table then you can, do the filtering by javascript but to pass the selection to flask route by javascript is another much harder thing. Also I would do selection on distinct values rather than all rows values.
I am currently rendering an simple HTML table using Flask/Pandas Dataframe (to_html) using rows from a table in an SQLite3 DB. 
How do I add checkboxes to the table?
After selections are made dump all the values from the Name column to a text file?
Example:

ID  Name    Manager 
1   server1 manager1
2   server2 manager2

Any help would be appreciated.
P.S. This is my first question on Stack overflow. If you need more information please let me know.