iPad Wi-Fi connection issues, including showing as connected with Wi-Fi disabled
I would do a factory restore to your iPad. 

Yes that does automatically update your iPad to iOS 11 but you can update to the lates iOS 10 update as well. The issue you have here is related to the software (and sometimes the connection with your other devices like an iPhone). 

Use http://ipsw.me/# as your source for iOS 10 updates for the iPad. Make sure that you backup your iPad if possible (iCloud Backup or iTunes) then download the file you need (choose it based on your exact device) and then save it. Go to iTunes and where it has restore iPad, press and hold on to the "Option" key on Mac (ALT on Windows) and click on restore. That should let you select the update file where you saved it and then use it to restore your iPad. 

If you need any assistance let me know.