waht i can tell is that its taking a while to switch from 3rd to 2nd. its normal to get a push when ur coming to a stop and then accelerate, because while ur stopping the car stays in 3rd gear (usually untill 10-20mph) and when u accelerate it has to switch to 2nd. thats where you feel the push.

your car will not down shift to 1st untill you have reached speed of below 5mph.

i dont know exactly why its lagging so much to down shift, but it could be your transmission control unit.

Swap that out before you decide to change your transmission its way cheaper and may work.

and the reason why it downshifts is to get more acceleration out of it.