Does łóżko refer more to the mattress or the bed frame?
usually both (together). In this sentence it is clearly a frame, when you buy an item -might be a frame, or a whole thing better check. A mattress without a frame in not really "łóżko".
Ona ma drewniany materac...
Doesn't accept "She has GOT a wooden bed" as a correct sentence.
That's an oversight, added.
"She has GOT…" anything is unaccepted as post-high-school English grammar and should be removed from answer set.
Sorry, what? Obviously "She has got" is a correct answer in British English.