how to check if linux mint is installed correctly

Answer 1

Arman as Martin Taylor mentioned pretty straight forward...
https://linuxmint-installation-guide.readthedocs.io/en/latest/

Answer 2

This may help you:
https://www.rootusers.com/check-which-linux-mint-version/#:~:text=From%20the%20Menu%2C%20select%20Preferences,of%20Linux%20Mint%20is%20installed.