Learn Russian: Волки живут далеко.

Even though wolves are animate (alive), волки is the subject of this sentence (they are the ones doing the living). The subject always takes the Nominative case. You only use the Accusative case (which would be волков for animate plural nouns) when the wolves are the direct object having an action done to them, like if you said "I see wolves" (Я вижу волков).

Why does живут have a в in it when the dictionary form is жить?

The verb жить (to live) is a bit irregular. In the present tense, a hidden в appears in all of its forms (like я живу, ты живёшь). Since волки is plural, we use the "they" form, which gives us живут.

Why does the plural noun волки end in -и instead of the usual -ы for masculine nouns?

The singular form is волк. Normally, to make a hard masculine noun plural, you add -ы. However, the Russian 7-letter spelling rule states that you can never write ы after the letter к (or after г, х, ж, ч, ш, щ). Therefore, you must use -и instead, making it волки.