Is the word "universe" supposed to be capitalized as a proper noun?
No, the noun "universe" is a common noun.
A common noun is only capitalized when it is the first word in a sentence.
The noun universe is a common noun as a general word for space and everything that exists in it, including the Earth and all the other planets; a general word for a particular sphere of activity, interest, or experience; a word for any universe of any kind.
When using the noun "universe" as a word for everything in existence , you may capitalize it as "The Universe" if you like, but it isn't required because it's not the name of a specific thing.