Considering this, can I have two versions of Office installed at the same time?
Microsoft does not recommend installing multiple versions of any Office product, but it is generally possible to install and use more than one version of Office on a single Windows computer. For example, you can install and use both Office 2003 and Office 2007 on the same computer.
Also Know, can you run Office 2007 and Office 365 on the same computer? Yes, you can run both on the same computer (I have that setup currently on my machine). Your current problem is with "Windows File Associations", it is not actually an Office problem. There is a simple fix, do an Office Repair of your Office 365 installation. That should reset the file associations as you wish.
Also to know is, are there two versions of Outlook?
Outlook 2016 for Windows and Mac
The two versions do share many of the same features, but there are key differences as well. You can use Outlook 2016 for Windows and Outlook 2016 for Mac if you're using a desktop or laptop computer. Outlook 2016 for Windows also works on most Windows tablets.
Can I have Office 2013 and Office 365 on the same computer?
You cannot have two Office 2013/Office 365 installation installed on the same computer. Once you purchase Office 365 Home Premium, you can use the same subscription with all of your PC's in your family.