Learn Turkish: Bu dükkân içki satıyor.

In Turkish, the present continuous tense (-ıyor) is very frequently used for general facts and habits, similar to the simple present tense in English ("sells"). While there is another tense specifically for general facts that you will learn later, using -ıyor here is perfectly natural.

Why isn't there an accusative case ending on içki? Shouldn't it be içkiyi?

We leave içki in its bare form because it is a general, indefinite object here. The shop sells "alcohol" in general, not a specific bottle. If you used the accusative case and said içkiyi satıyor, it would mean "it is selling the alcohol" (a specific one we already mentioned).

Why does dükkân have a little hat over the letter 'a'?

That mark is called a circumflex. It tells you to pronounce the 'k' before it softly (similar to the 'k' sound in "cute") and slightly lengthen the 'a'. It is used in certain words borrowed from Arabic or Persian to keep their original pronunciation.

This lesson is about the ablative case (-DAn). How would I use it to say I am buying alcohol from this shop?

You would add the ablative suffix to the shop itself: Bu dükkândan içki alıyorum (I am buying alcohol from this shop). Adding -dan changes "this shop" into "from this shop".