After engine is warmed up let it idle and unplug the IAC(idle air control) valve wires, engine RPMs should drop to 500 or engine may even stall, either is good it means no vacuum leak.
If idle stays above 700 then there is a leak.
Leaking EGR valve will cause rough idle, by allowing exhaust gases into intake when it shouldn't.
In general it does read like a spark issue, idle has a richer mix on purpose, and when you push down on gas pedal quickly that richens it even more.
Weaker spark can't ignite rich mix.
I would pull out 1 spark plug from each cylinder and have a look at the tips to see what they tell you about conditions in each cylinder.
If you could do a compression test at this time, 1 spark plug remove from each cylinder, it could tell you if one cylinder has lower compression than the others.
Low compression in a cylinder has the same symptom, compression = heat, if fuel is not heated up enough then spark plug can't ignite it.
As you increase RPMs compression increases so cylinder starts to fire.