Is kaffe an ett- or an en-word?
Confusingly, both. It depends on what you mean.
En kaffe means a cup of coffee. "Vill du ta en kaffe efter jobbet?" = "Do you want to go grab a coffee after work?"
Ett kaffe refers to the beverage itself, or to a certain type of coffee.
I struggle with hearing the pronounced difference between "han" and "hon". They sound exactly the same to me in the audio clips I'm given... Any tips?
I'm having trouble differentiating between "drinks" and "is drinking", among other vebs. How can I tell the difference?
Usually there's no need to worry about that. Swedish seldom bothers with both forms. As is discussed elsewhere.