Is DPS in sorted movelists including STAB damage?

No, it doesn't appear to take STAB into account. However since STAB is dependent on what type of Pokemon uses the attack, it wouldn't make sense to include STAB on the Moves List.

What I would like to see is STAB applied when looking at a specific Pokemon. For example, http://pokemongo.gamepress.gg/pokemon/1, instead of Bulbasaur's Vine Whip attack being 15.38 DPS, it should read as 19.23 DPS because of STAB.

That's a very good answer by LoneAsp, but what should be displayed is instead "15.38 + 3.85 STAB", meaning the native amount and added amount can and should very easily be differentiated.