Learn Russian: Где твой пропуск? Я забыл его дома.

No, but it is a great catch! Here, дома is an adverb meaning "at home" and is stressed on the first syllable (дОма). The plural word for "houses", which you are learning in this lesson, is spelled the same but stressed on the last syllable: домА.

What is the plural of пропуск (pass)? Does it follow the rule from this lesson?

Yes! When пропуск means a "pass" or "permit", its plural takes a stressed -а: пропуска. (Note: if it means an "absence" or "skipped class", the plural is actually пропуски, but пропуска is what you will use for ID badges and security passes).

Why do we use его here if a pass is an "it"?

In Russian, pronouns must match the gender of the noun they replace. Since пропуск is a masculine noun, it is treated as "he" (он). Because the pass is the direct object being forgotten, we use its accusative form, его.

Can we tell if the person speaking is male or female?

Yes, the speaker is male. We know this because the past tense verb забыл (forgot) is in the masculine form. A female speaker would say Я забыла.