in terms of getting up or not, it seemed pretty clear on the replay that Dzeko took a knee to the thigh/quad, which isn't a big injury but certainly hurts in the immediate term. Could explain why he was OK to play but stayed down a minute or so, anyone who has taken one of these can relate
My opinion is more nuanced. I think Liverpool were OK and correct to attack as they did, but probably should have played it out after Garcia's half clear. At that point, City had beaten the initial attack and it seemed like the advantage Liverpool were not wanting to squander due to injury had been lost. I think you'll see teams frequently press the advantage and then play it out if the advantage is lost, but here they made a great pass back into damage and truthfully benefit from a great strike from Sturridge.