Learn Finnish: Minä olen kaupassa ja ostan lahjan. - I am in the shop and I will buy a gift.

This is the genitive-accusative ending that you are learning in this lesson! Adding -n makes lahjan a total object. It shows that you are buying one whole gift and the action will be fully completed.

Why does the English translation say 'I will buy' if Finnish doesn't have a future tense?

You are right that Finnish uses the present tense for both present and future actions! The clue here is actually the total object lahjan. Because the object has the -n ending for a completed action, it tells us that the buying is a specific event that will get finished. In this context, it naturally translates to a future action like 'I will buy' in English.

The dictionary word for shop is kauppa. Why does it only have one p in kaupassa?

This is a quick refresher on consonant gradation. When you add a closed ending like the 'in' ending -ssa, the strong double consonant pp in the basic word kauppa weakens into a single p.

Do I have to say minä at the beginning, or could I just start with olen?

You can absolutely drop minä. The -n ending on the verbs olen (I am) and ostan (I buy) already tells the listener exactly who is doing the action. Keeping minä in the sentence just makes it sound a bit more emphatic.