Learn French: Leur client ne comprend pas le problème. - Their client does not understand the problem.

In French, possessive adjectives match the thing being owned, not the owners. Even though 'their' refers to multiple people, they only have one client (singular). If they had multiple clients, it would be leurs clients.

What subject pronoun does Leur client count as for the verb comprend?

Leur client counts as a third-person singular subject, just like il (he) or elle (she). That is why the verb is conjugated as comprend.

Can I translate 'does not understand' as 'ne fait pas comprendre' since 'fait' means 'does'?

No, French does not use a helper word like 'does' or 'do' to make negatives. You simply put ne and pas directly around the main verb. 'Does not understand' is just ne comprend pas.

The word problème ends in an 'e'. Why does it use the masculine le instead of la?

While many French words ending in 'e' are feminine, there are exceptions. Words ending in '-ème' are usually masculine. Le problème, le système, and le poème are common masculine exceptions you just have to memorize!