Learn Danish: Hun ligger på sofaen under et tæppe. - She is lying on the sofa under a blanket.

Ligger means "lies" or "is lying" (resting in a horizontal position). Lægger means "lays" or "is putting something down" (an action). Because she is already resting on the sofa, we use ligger.

Why is the English translation "is lying" when the Danish sentence just says ligger?

Danish does not have a present continuous tense like the English "is + -ing" form. The present tense verb ligger translates to both "lies" and "is lying". You have to rely on context to know which sounds best in English.

Doesn't et tæppe usually mean a carpet or a rug?

Yes, but it also means a blanket! Et tæppe is a very versatile word in Danish. If it is on the floor, it is a rug or carpet. If it is on a bed or a sofa, it is a blanket.

We use på (on) for the sofa. Do we use på for all types of seating?

Not always. Just like in English, you sit or lie på sofaen (on the sofa) and på stolen (on the chair). But if you are in an armchair, you would use i (i lænestolen), because you sink into it.