Hi Steve,
For the first question, I would like to confirm if you can send the messages from this mailbox when you login to Office 365 OWA on the computer, not tablet.
If you can't, it means that you don’t have “send” permission from this mailbox. You need to contact your administrator to grant the proper permission to you.
If you can, the issue might cause by some configuration issue on the device.
Please check them referring to the link below:
http://help.outlook.com/en-us/140/dd936215.aspx
For the second question, you can sync Office 365 Emails with Outlook on your PC. Please connect Outlook to Office 365 Account referring to the link below:
http://help.outlook.com/en-us/140/ms.exch.ecp.useoutlookanywhere.aspx
Thanks,
Edward Qu