If I close this Microsoft account will my Xbox account that uses this gmail close and erase...

Hi there. My name is Ron, I'm an independent advisor. I will be more than happy to help you.

It deletes every service related to a Microsoft account including any data from Microsoft Xbox.

If it's a Microsoft e-mail (Hotmail, Live, Outlook) then you can't, they are automatically WLIDs. If it's a non-Microsoft e-mail (Gmail, Comcast, etc.) then you can go to account.live.com and click on Close account, that will close the Windows Live ID.
Keep in mind that that will break your Xbox account if that is the WLID the Xbox account uses.

Yes, the Gmail account is still active it's just the Microsoft account that got closed because of inactivity.

Hi reytorres05,

Thanks for your post. If you are using a 3rd party email as your Microsoft account alias and you delete the Microsoft account it is on, you will not lose your email. Since Google manages Gmail, Microsoft is unable to close your Gmail account.

I hope this helps.

If a Microsoft Account has an alias from a outside email provider (e.g. Gmail), and you remove that alias, it won't close the outside account that hosts that email address.

For example, my Microsoft Account has an @gmail.com currently configured as an alias. If I remove that alias, I'll still be able to access my Gmail via Gmail, using my Google Account while I continue to access my @outlook.com addresses via Outlook.com using my Microsoft Account.

Mind you, this is just general advice. If your mum has been using her Microsoft Account and Outlook.com to sync all her mail from Gmail she may not be familiar with Gmail's interfaces (and may not even have all the mail in her Gmail inbox anymore if it was sync'd to Outlook.com and deleted). You should also make sure your mum doesn't use the Microsoft Account to access any other services.

- Jason