Genau. I find http://de.wiktionary.org/wiki/Wiktionary:Hauptseite very helpful. There's a box in the upper right corner that states each case in singular and plural forms. I hope it helps you and anyone else who decide to use it.
Yes; the vowels are pronounced differently.
Berge is [ˈbɛɐ̯ɡə] and Bürger is [ˈbʏɐ̯ɡɐ].
The first vowel has [ɛɐ̯] (starting with an unrounded open-mid vowel) versus [ʏɐ̯] (starting with a rounded close-mid vowel); the final vowel has [ə] (a central mid "neutral" vowel) versus [ɐ] (a lowered central vowel).
Even Bürge and Bürger are pronounced differently -- the central vowel at the end is at different heights.
The distinction are subtle and may be hard to capture as they are not found in English, but they are different.
I do Duolingo on a PC, and there are indeed oral exercises that show no written text. I came to the comments sections precisely because I was curious about using "Sie" rather than "sie" which I had already typed in. I have suspected that the lessons are somewhat different on smartphones, but this is a really big difference that could cause a lot of confusion when discussing possible answers.