Learn Farsi: هر رده ده درصد از تمام جمعیت را شامل می‌شود. - Each category includes ten percent of the entire population.

To say a percentage of a whole, you use az (from/of) after the percentage. So "ten percent of the entire population" is dah darsad az tamām-e jam'iyat. We don't use the ezafe (which would look like dah darsad-e...) because az is specifically required to show the portion coming out of a larger group.

Why is the direct object marker rā placed all the way after jam'iyat instead of right after the percentage?

The entire phrase dah darsad az tamām-e jam'iyat acts as one big noun phrase receiving the action. The marker rā always attaches to the very end of the complete direct object, no matter how long the phrase is.

The subject here is har rade (each category), so the verb is singular. If the percentage itself were the subject, how would verb agreement work?

If a percentage phrase like "ten percent of the people" is the subject, the verb agreement usually follows the logical subject. Since "people" is plural, the verb would typically be plural, even though the word "percent" itself is singular.

I see shāmel mishavad at the end. How does the verb shāmel shodan (to include) connect to the rest of the sentence?

shāmel shodan is a compound verb meaning "to include" or "to comprise". The thing that is included (in this case, the ten percent) is the direct object. Because we are talking about a specific ten percent (the ten percent of this specific population), we use rā right before the verb.