Access 2007 and 2010 won't run in 2013, when you try to open the database it gives an error message saying something like "can not open a previous version"
I run 20903 mde, 2007/2010 create accde in Access 2013 without any issues.
From the error I suspect hat you have renamed the .accdb/.accde to a .accdr. Is that correct? If yes, try renaming back to a .accde or .accdr. Does it run after renaming?
My boss wants us to upgrade Access on our terminal server to Office 365, however it doesn't support the current version we are using which is 2010.
There may be a licensing issue with the keys from the Office 365 plan when installing it on a terminal server. Normally you purchase Office 2013 Professional Plus (exact same thing as with Office 365) but through a Volume Licening. I would check with the vendor you used to get Office 2010 for the terminal server.
Office 365 is just a subscription plan to a suite of services, which usually includes Office Professional Plus for the desktop.
With all the Office 365 plans that include the desktop apps you get Office Professional Plus the full office suite). Depending on the plan how many licensees a user gets to install on their multiple devices, not multiple users.
What Office 365 subscription plans do provide is other hosted services that you may not need.
To get
Office 2013 Professional Plus you have to options.
1) Regular retail version - single copy or Volume License (VL)s.
2) Office 365 subscription (which has a license to install on 3 to 5 of the user's devices (laptop, desktop, etc) depending on the plan.)
Either way you get the exact same desktop software. The only difference is the license type.
It appears what people are getting confused when they hear Office 35 when related to access as meaning web database. Unless you have a sharepoint server (could be through Office 365 then a web database becomes an option. But is is definitely not a requirement with Office 2013 through a from Office 365 subscription.
For small business with less than 10 computers and home users the most cost effective way to get Office is through the Office 365 subscription. It give small business the same centerpiece level computing at a very low cost.