You need Excel services to show data. It doesn't matter that you're using the 2010 version for masterpage as it shouldn't really effect it!
You could try one thing though, by setting the 2013 site as 2010. You will still run under the same issue though! The webpart is still the same.
I have a strong feeling even though not tested is that you need to have excel services running before you can see those custom configurations. I presume if it's not active than you should have the option to set something that would break the webpart. When you save the webpart properties those custom parts will communicate with the Excel service and if it's not available than it wil fail.
Here is a quick read which I should have done but it proves that I'm correct is this statement.
Excel Services enables these commands automatically. To disable one or
more of the commands, clear the check box for that command.
It was hidden and should have been highlighted like above! But that is your answer, the Excel service is your issue.
You should have set up all service that you require before you migrate the contents over! Now you might get complications that even though you activate and install Excel services it still might not appear!
I would test to make sure that the above is correct by adding another excel webpart to page and see if the properties appear. Then add the excel service and check the webpart again to edit the properties. You should see the properties - check with 2010 migrated webpart if it's ok; if not check with 2013 webpart.