You have done some great comparisons but the devices compared do not belong to the same league. 820 is not in that league, you can compare GS4 with 920 or 1020. But comparing 820 with GS4 is not simply not fair.
Sent from my Nokia-Lumia-820 using Tapatalk
Well , More fair comparison would be Galaxy S2 (2011)
Both have 4.3 inch AMOLED displays , 480 x 800 resolution , 8MP camera which can shoot 1080p and 1GB RAM.
I didn't use Galaxy S2 as my phone but spent lot of time with the device since its lying around the house.
In terms of Photos , Lumia 820 triumphs but on video , Lumia 820 suffered some stutters and little bit focusing problems while GS2 performed quite well.Microphone quality is pretty much equal.
On , performance however , Lumia 820 is a beast and shortcut camera key comes handy all the time.Burst shot wasn't available on Galaxy S2
Note : Regarding stutter :
I got one "video cut" for every 28-29 seconds If I put "store on SD-card option" , I was using only sandisk class 6 card on Galaxy S2 and Sandisk Class 10 Ultra on Lumia 820 but GS2 didn't have any problems.I used same card on Galaxy S4 , didn't experience any stutters.
So , only problems I had with 820 was focusing problem on video (aside from stutter which occur due to WP problem with external storage or software issue)
Galaxy S2 has bad battery life and also performs plainly bad in Jellybean version (Gingerbread was so smooth).
GS2 screen also seems little bit more pixelly than 820.I think it has to do with WP8 optimization of fonts and visuals.