Hi Chris,
To verify if the external user has been invited to your site, please click the Site Actions > Site Settings, and open the People and Groups page. If the Hotmail user has been invited successfully, you should be able to see the user display name in the People and Groups page. If the external user is listed there, please check the permissions for the user to see if proper permission has been assigned, otherwise the external user may not have been added to your site.
To try to invite other Office 365 users, you can sign up a trial account and create some test users in the trial account. Meanwhile, please try to invite an Hotmail user in the top level site collection instead of your Team Site. To access the site collection, please click the Edit website in MOP.
Thanks,
Reken Liu