How do I uninstall the Apple Mail app and reinstall it?
To uninstall try with AppCleaner, it's free in the AppStore.
Apple Mail is a system app so, in AppCleaner, try double click on the checkbox and then proceed to Delete the app.

For Reinstall.
Apparently you'll need to do a reinstallation of OSX.
According to this Apple page, a reinstallation wont affect your files, can't tell for myself, I've never done that. 
Simply make a backup before anything and may the force be with you.

OSX Reinstallation


Reboot your Mac and hold the COMMAND + R keys while rebooting.
You will enter to Recovery environment.
Choose reinstall.