What you want to do is easy.
On the phone go to settings, accounts, your email and accounts. The square with the big +, add an account, Google will be one of the selections. There is where you add your Google account.
Adding this account will bring your gmail email, contacts and calendar to the phone. You can select if you want to sync these individually or not sync at all.
By the way, the domain is the - email name @ domain . com
So the domain of your Google account is - gmail
Need to make sure of this, Important. Answer this one.
Do you have an account from Microsoft on the phone?? This should be a live, Hotmail or outlook.com account? For everything to work together this is needed.
What account is displayed at the top of the email and accounts page???
This may be of help in the future - Need help with Windows 10 Mobile?
http://windows.microsoft.com/en-us/windows-10/support-mobile?ocid=wpcom_nav