Rough idle at cold start . Is it injector, thermostat, vapor purge canister, gasket, vacuum leak? MAF and 4 igniters were swap with another corolla that was supposedly running fine.

Yes, I have check the plugs. Discover
that the fuel pump unit was missing
necessary O rings to keep the unit
taut under working pressure. So the
car seems to run fine, after changing
out duel filter and strainer.  SAVE  
one thing, well maybe two things.  
The car starts now, but at cold start  I
needed 2-3 cranks to get it to start --
I may have damaged the stsrter?  
The cold start has a rough idle or the
engine lopes it seems.  A few pump
of the gas, the engine will surge to
higher RPM and stay around or
above 1000RPM til warms up but will
drop down to 750RPM or so after 5-
10min. I drove the car around for
several miles, has good acceleration
and idles seemingly normal at 700-
750RPM, engine light came on.
Checked the code, 0171 lean bank1.  
300 random fire.  302,304 misfire.  
Let car cool. Disconnect battery.
Reconnect after few min, car Rough
idle start 400-600RPM.  Try to
maintain idle by pumping gas pedal.  
Car warmed up but rough idle
remains.  Hesitate when try to
accelerate or back up.  10-15mph
max.  Drove around block and
parked. Stop car. No codes showed
up.  Disconnect battery a few min,
reconnect.  Crank engine 2 times, car
start weakly but idle good at
750RPM.  Car Drove fine.  Code
0171, lean bank1.