ok, so when you had a dead battery, your car didnt crank at all? because, lets all be on the hopeful side, lets say that the whitesmoke was unburnt gas, but since you had a dead battery.. well it probably was coolant
now, about the delay in pushing the gas, lets hope this is just when you started your car b/4 it warms up, because who knows, the tranny fluid can take some time to get to the torque converter
and the coolant in the oil, does that mean your coolant is brown? that sucks dude, how did you all of a sudden crack your head, or any of the sleeves/pipes or whatever you call them that go through the engine crack? in 1 day? not likely