Del vs. El
One of the questions was, "El gato duerme cerca del perro." Wouldn't this be, "The cat sleeps near of the dog"? How is this correct?
If you directly translated each word individually, that would be true. I think "de" needs to follow "cerca" before you add a noun... So then you can look at "cerca de" as a phrase that takes on the meaning "near to".
Well you can't translate according to what the words mean, but by how they're used. When you want to say something is "close to" another thing, you should say "cerca de," and not worry about what the dictionary tells you the words mean individually. The word "del" is just a combination of "de" and "el." So it could be used to say close "to the" and also in a possessive sense, "of the": "That's the teacher's wallet," "Esa es la billetera del profesor."