Not necessarily, though "our lives" works too. "Our life" can refer to the collective experience of a group of people, where "our lives" would more strongly individuate those experiences. "Our lives" can also refer to our literal lives, in the life and death sense, but "our life" would not work there.
"Жизни", because "ь" changes to "и" in plural, to keep the preceding consonant palatalised.
That's what I thought too. I know it's 'life', but then seeing наша made me write lives instead.
Meets B.J. Blazkowicz. «Моя жизнь!»
Just Google that name if you do not understand the reference.
Agreed. Its almost like she is putting another vowel in front of the Ж. In general I often have trouble with the duolingo Russian audio, but this could be due to my untrained ears.