Hi PBD,
As I understand, you have set up Office 365 account through Outlook client on two computers. One of you computers can pull emails and update emails correctly, however, another computer can’t receive new emails. Am I right?
It’s ok to host your domain at other provider and use Office 365 services, generally, this will not cause any problems. In your situation, the problem should be caused the Outlook connection issue on your second computer.
Regarding to the problem, at first, I’d like to clarify the following details:
1, Can you send and receive emails with OWA (Outlook Web App) properly?
2, Have you tried to click on Send/Receive on your second computer, does any error occurs?
If error occurs when you click on Send/Receive icon in Outlook client, then, please try to fix the connection by doing the following:
In Outlook client, click File, click Account Settings and select Account settings; select your Office 365 account, click Repair, follow the wizard to try fix the issue.
If the error still persists, I suggest try to reconfigure your Office 365 account in Outlook client on your second computer to see if the problem can be fixed. You may refer to Connect Outlook to This Account.
For question two:
To invite external users to SharePoint Online, you can refer to Share a site with external users. To have specify external user to view specify information, you can do this by break the inherit of the website and then invite the user at the specify website.
Thanks,
Evan Zhang