Learn Finnish: Me kirjoitamme kotona. - We are writing at home.

This is because of consonant gradation. Kirjoittaa is a Type 1 verb. To find the stem, we remove the final -a, leaving kirjoitta-. When we add the -mme ending for "we", the syllable becomes closed (ending in a consonant). This triggers a "weak" grade, so the double tt weakens to a single t, giving us kirjoitamme.

Does kirjoitamme mean "we write" or "we are writing"?

It means both! Finnish does not have a separate continuous tense like English does. Depending on the context, Me kirjoitamme can be translated as "We write" (in general) or "We are writing" (right now).

Do I have to say Me kirjoitamme, or can I just drop the pronoun and say Kirjoitamme?

You can just say Kirjoitamme. Because the -mme ending clearly tells us the subject is "we", the pronoun Me is optional. In spoken Finnish, people often include it, but dropping it is perfectly correct.

Why is "at home" kotona? I thought "in" or "at" usually ends in -ssa or -lla.