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This thread is resolved. Here is a description of the problem and solution.
Problem: I want to translate only my theme from English to Mongolian, not the site's content.
Solution: You would need to configure 2 languages (English and Mongolian), set the Mongolian as default language and then translate the theme via String Translation.
Relevant Documentation: https://wpml.org/documentation/getting-started-guide/theme-localization/