Learn German: Wir brauchen jetzt das Auto.

Because the subject is wir (we). When using the plural pronoun wir, regular verbs take the -en ending. This means the conjugated verb looks exactly the same as its base dictionary form!

How would the verb change if the subject was ihr (you all) instead of wir (we)?

Since brauchen is a regular verb, you would take the stem (brauch-) and add the -t ending for the plural pronoun ihr. The sentence would become: Ihr braucht jetzt das Auto.

Why is jetzt (now) placed before das Auto? In English, we usually put "now" at the very end of the sentence.

German word order is a little different. Time words like jetzt generally come right after the verb and before the object (das Auto). A good rule of thumb in German is that time information prefers to come early in the sentence!

das Auto is the object being needed. Shouldn't its "the" word change because it is in the accusative case?

It actually is in the accusative case, but the word das doesn't visibly change! In German, only masculine nouns (words that use der) change their "the" word to den when they are the direct object. Neuter words like das Auto stay exactly the same.