Learn Russian: Она берёт газету.

The rule that inanimate objects stay the same only applies to masculine and neuter nouns! Feminine nouns ending in -а (like газета) always change their ending to -у when they are the direct object, regardless of whether they are alive or not.

The dictionary form for "to take" is брать. Why does the word look like берёт here?

The verb брать has an irregular stem in the present tense. Instead of keeping the -ра-, it changes to бер-. Because the subject is она (she), we add the matching ending -ёт, making it берёт.

Is the letter ё in берёт pronounced like an English "e"?

No, the letter ё sounds like "yo". Another helpful rule for beginners: whenever a Russian word has the letter ё, that syllable always gets the vocal stress!

There is no word for "the" in the Russian sentence. How do we know it means "the newspaper" and not "a newspaper"?

Russian completely lacks articles like "the" or "a". Она берёт газету can translate to either "She takes the newspaper" or "She takes a newspaper". You just use the context of the conversation to figure out which one makes sense.