I recently went through the process of updating my state and country fields to picklists using the standard Salesforce tools. Perhaps foolishly I completed this update directly in production. However this update--which forces a value for [Country] on new object creation (Lead, Account, etc.) has caused a number of errors in the Apex test classes in my org which didn't have a [Country] field because it wasn't required before this update. Normally I would just make edits to the test classes and call it good but most of the Apex in my org is in the form of managed packages which I can't edit. So now I'm in a position where my code coverage isn't high enough to import new code AND I can't edit the existing code to bring the code coverage above 75% because I don't have access to the managed packages. Any suggestions on where to go from here?
Managed Package code coverage should not affect your own deployment of code in your org's namespace: Unit Test Code Coverage and Managed Packages
You should be able to bring your own code coverage up to above 75% without manage package test classes passing.
It's tough to tell from your post why the tests are failing, but I assume it's because they're inserting state values without country values, going by this:
However this update--which forces a value for [Country] on new object creation (Lead, Account, etc.)
Country values are only required if you're also setting a state value. If the code causing errors is only using states from one country, one workaround would be to set the org's default country to that country.
If the tests are failing because they're inserting unrecognized state and country values into the text fields, you could also add those values to your state and country picklist configuration and uncheck the "visible" option. Invisible state and country picklist values are valid for API inserts, but do not appear in the end-user UI picklists.
I also want to clarify that turning on State and Country Picklists is reversible. You might have to undo updates to metadata like reports and formulas that use the picklists, but turning on State and Country Picklists is not a permanent change.