Why isn't "stare" in the instrumental here? Is that only for nouns after być?
Not sure if that's helpful, but you're not the only one. Apparently those are very similar sounds for the learners, at least in the beginning.
Personally I never considered them similar before I started dealing with Duolingo... but well, such things are obvious to the native ear.
zwierzę...i can hear a slight difference with "e" of stare...but "ę" should not have a "n" sound in the end?
Not really. It's just nasalized "e". For some people it is helpful to approximate it to 'en', but that's really too much.
And in fact if ę is the final sound of the word, it sounds almost the same (or exactly the same) as the normal e.