Sounds like a blown head gasket. check the oil if it looks like chocolate milk you defiantly have blown gasket. The white smoke is because antifreeze is getting into the oil.
If the belt you just had changed was the timing belt it may be related. I loose timing belt will defiantly cause a problem. Jumpstarting is not going to solve anything...unless the battery is run down. Check back with whoever did the work and ask them if they did anything that might have caused this.
generic wiring schematics seldom match the OEM wiring as to get the diagram the book publishers have to virtually strip down a vehicle as the diagrams are copy write protected. IN many cases the manufacturers of the vehicles change wiring harness suppliers and the OEM schematics will vary to your vehicle. When in doubt take it to an installer and have them fit it as diy people generally install incorrectly and produce faults in the CPU program
Do not let this car over heat a blown head gasket will ensue. Locate and replace the fan relay, if that doesn't fix the problem unplug fan and use jumper wires to ground the plug then feed power to the positive side of plug to check fan, if that doesn't work replace the coolant temperature sensor located somewhere near the intake manifold or thermostat housing area.
Disconnect the negative and positive battery cables from the terminals on the battery. Take that battery out. Now then buy another battery for you make and model. Put the positive and negative battery cables on the battery terminals of the new battery. MAKING SURE, you have them on the RIGHT CABLES ON THE RIGHT TERMINALS. POSITIVE --- (RED) as the NEGATIVE CABLE is ----(BLACK). If you get them mixed up and try to start the car or something. You will blow a ton of fuses. and no telling what else! Its not hard as long as you can read the (-) << NEGATIVE (BLACK) WIRE and (+) << POSITIVE --(RED) WIRE on the battery.
This sounds like a transmission problem. I would start by checking the transmission fluid. You must start the engine and put the car in Neutral before checking the transmission fluid. I would bet that it is low on transmission fluid.
brand new batteries have a warranty because they fail early on in life. Brand new should equal quality but sadly it does not. It is good to see that the mechanic is interested enough to follow up on his work so hang in there as he may get to the root of the problem
Check the spline nut that is at the base of the wiper. It is under
little plastic flap at the base of the wiper. Flip it open and tighten
the nut down and make sure both wipers are in the down position when
you do it.
PO732 - Gear 2 incorrect ratio. PO33 - Turbo/supercharge Bypass Valve control circuit/open. PO741 - Torque converter clutch solinoid stuck off. PO783 - 3-4 shift. You have a big problem with your trany.
your going to need to replace the igniton or the electrical switch part in the back of switch may have broke apart and jammed it,take it apart and seeif the key turns once the switch part is off the key part,(electric ignition switch is removed from the cylinder lock).