Learn Farsi: طراح داشت گریم را چک می‌کرد.

To say someone "was doing" something in Farsi, we use the past continuous (here, چک می‌کرد) and add the conjugated past tense of داشتن (to have) as a helper verb. Since "the designer" is third-person singular, the helper is داشت and the main verb is می‌کرد.

Why are داشت and چک می‌کرد separated by گریم را?

In Farsi's past progressive structure, the helper verb (داشت) usually comes before the direct object, while the main verb (چک می‌کرد) stays at the very end of the sentence. So the pattern here is: [Helper] + [Object] + [Main Verb].

Why does the prefix می come in the middle of چک می‌کرد?

چک کردن (to check) is a compound verb made of an English loanword (چک) and a light verb (کردن). The continuous prefix می is always attached to the verb part, so it goes right before کرد, not at the beginning of the whole phrase.

I've learned آرایش for "makeup" before. How is گریم different?

آرایش refers to everyday cosmetic makeup. گریم, borrowed from French, refers specifically to theatrical, film, or special effects makeup. Since the sentence mentions a "designer" (طراح), they are likely working on a stage or film set.