OBD-II Trouble Code: P0300 Random/Multiple Cylinder Misfire Detected

worth reading the symptoms describes the issues that you are having which link nicely into the p1781 code information below.

P0700 Transmission Control System TCS Malfunction

doing a little reserch into this i take it that your car is auto? if so the reserch that i am coming up with is saying your throtal body positioning sensor is the issue perhaps


http://www-odi.nhtsa.dot.gov/owners/SearchVehicles fill in the details for our car and this is the information that comes up that i found relating to the error code. may be esiest after entering the details to run a find on page search going through the pages one at a time.

Service Bulletin No.: 050601030
Component(s): ENGINE AND ENGINE COOLING

NHTSA ID Number: 10017820

All Products Associated with this Service Bulletin

Details

0 Associated Documents

Manufacturer: General Motors LLC
SUMMARY:

1.6L L91 INTERMITTENT MISFIRE COLD DURING DECELERATION OR STEADY STATE HIGHWAY DRIVING AFTER COLD START WITH DTC'S P0300, P0700, P1781 AND/OR U0401 SET (REPLACE VALVES AND VALVE STEM SEALS) *TT UPDATED 11/02/07 *NJ
so in conclusion possibly one of two or both things.
1: bent exhaust valves (if my reserch is correct these will be the damaged valves) could have been damaged when the car over heated.

2: throtal body positioning sesnsor is not reading corectly would cause rough idle and transmision issues as the engine doesn't know exactly whats going on at this point confusing it.

i personaly would lean towards the bent valves maybe have a compresion test done just in case it is this.