Learn German: Wir leben ohne Angst. - We live without fear.

The word ohne (without) always requires the accusative case. However, we are using the noun Angst (fear) on its own here, without an article like "the" or "a". Because there is no article to change, you don't see any visible accusative markers in this sentence.

What if I wanted to say "without the fear"? Would the word for "the" change to show the accusative case?

It would be ohne die Angst. The word Angst is a feminine noun. In the accusative case, the feminine article for "the" remains die (exactly the same as the basic subject form). You only see an obvious spelling change after ohne when you use a masculine noun, like ohne den Hund (without the dog).

Why is Angst capitalized? Is it to emphasize the emotion?

No, Angst is capitalized simply because it is a noun. In German, every single noun must be capitalized, whether it is a physical object you can touch or an abstract concept like fear or love.

Does the verb leben have a special ending here?

The verb leben (to live) is completely regular. Whenever the subject is Wir (we), verbs take their standard infinitive form, which almost always ends in -en.