Learn German: Wir kaufen unsere Wohnung. - We are buying our apartment.

Because Wohnung is a feminine noun. Whenever a possessive word like unser (our) or sein (his) goes before a feminine noun in the nominative or accusative case, you must add an -e to the end.

Is unsere Wohnung the subject or the direct object here?

It is the direct object (the accusative case), because the apartment is the thing being bought. Luckily, for feminine nouns, the accusative ending is the exact same as the nominative (subject) ending: they both just take the -e on unsere.

Is there a rule to help me remember the gender of Wohnung?

Yes! As a general rule, German nouns ending in -ung are always feminine. Knowing this trick makes it much easier to remember that you need feminine endings like unsere or ihre when talking about an apartment.

The English translation says "are buying". Why isn't there a word for "are" in the German sentence?

German doesn't have an "-ing" (continuous) tense like English does. The single word kaufen can mean either "buy" or "are buying" depending on the context. You never need an extra word for "are" when saying you are doing something in German.