Learn Russian: Где мой паспорт?

In Russian, the present tense of the verb "to be" (am, is, are) is almost always skipped. So, "Where is my passport?" simply becomes Где мой паспорт? (literally "Where my passport?").

This lesson teaches у меня for possession. Why don't we use it here?

The phrase у меня есть means "I have" (literally "by me there is"). Here, we just need the possessive word for "my", which is мой. Use у меня есть to state that you own something, but use мой to simply describe an object as "my [object]".

How do we know to use мой instead of моя or моё?

The word мой changes to match the gender of the noun it describes. Паспорт ends in a consonant (т), meaning it is a masculine noun. Because of this, we must use the masculine form мой.

Could I change the word order and ask Мой паспорт где?

Yes, Russian word order is flexible. However, putting the question word Где at the beginning is the standard, most natural way to ask this. Putting где at the end sounds more like you are expressing surprise: "My passport is WHERE?".