If you keep getting prompted to log into Office 365, try to keep your Windows applications, such as Word or Excel open after the first prompt. Also if you log in through the Microsoft login page make sure you check the "Keep me signed in" checkbox.
Check to see if you are using Microsoft Office 2010 and Windows OS 7. If your computer's configuration is not up to date, you may experience issues with SharePoint.
There is a per file limit of 10 GB. There is a 1 TB limit per site collection. Site collections are large groupings like Central Admin or FAS, as well as some dedicated site collections. Separate site collections are also issued for HRCI data collaboration.
Each site collection can have up to 2,000 subsites and they would all share the site collection quota, so there is no per subsite limit.
Yes. SharePoint users can send each other embedded links to SharePoint documents through their Harvard email. Unlike an email attachment, embedded links are more secure because they can only be accessed by someone who has the appropriate permissions.
Not necessarily, and may actually complement these other tools. It may be that you find certain tasks easier or more convenient in SharePoint, and so decide to use it for those areas. Either way, you will decide which Harvard tool works best for each of your administrative and collaboration needs, and you are free to use that tool as needed. Your department may have specific recommendations or requirements to utilize SharePoint for a particular purpose or project, but SharePoint is not mandated in any way.