Learn Finnish: Hän kirjoittaa kirjeen itse. - He writes the letter himself.

Kirjeen is the object of the sentence (the thing being written). When a regular singular noun is the total object of an action, it takes the accusative ending -n. Notice how this is different from the personal pronouns in your current lesson, which take a special -t ending for their object form.

If the object was a personal pronoun instead of a letter, would it also end in -n?

No, and this is exactly why personal pronouns are special! Regular nouns like kirje use the -n ending for objects, but personal pronouns use their own unique accusative ending: -t. For example, if you wanted to say "He chooses you", it would be Hän valitsee sinut (using the -t ending), not sinun.

How does itse work here? Does it change depending on who is doing the action?

Itse translates to "oneself" (myself, yourself, himself, etc.). Here, it acts as an emphasis word meaning "himself." It doesn't change form based on the person. If you wanted to say "I write the letter myself," the word stays exactly the same: Minä kirjoitan kirjeen itse.

Why is there a double vowel at the end of kirjoittaa?

To make the hän (he/she) form in the present tense, Finnish verbs usually double the last vowel of the verb stem. The stem for "to write" is kirjoitta-, so we add an extra a to get kirjoittaa.