Questions & Answers about У нас есть новый матрас.
Since this lesson is about verbs like 'ложиться' (to lie down), how would I say "I am lying down on the new mattress"?
You would say Я ложусь на новый матрас. Remember that verbs of assuming a position take на or в plus the accusative case to show the destination. Since матрас is an inanimate masculine noun, its accusative form looks exactly like its nominative form!
Why isn't матрас in the accusative case here, since it's the thing we "have"?
In Russian, possession is expressed with the structure У + [Genitive] + есть + [Nominative]. So literally, this sentence means "By us, there is a new mattress." Because the mattress is the grammatical subject of the Russian sentence, новый матрас stays in the nominative case.
Is it spelled матрас or матрац? I feel like I've seen both.
Both матрас and матрац are strictly correct and mean the exact same thing, but матрас is much more common in modern everyday Russian. You can just stick to using матрас.