The part they did not replace could be the problem. That's the accumulator, it makes the fuel pressure even. If it fails, it can cause a loss of fuel pressure when the car sits for too long, which sounds like what may be going on. I assume the weather is warm there. in cold weather, the incorrect ie platinum sparkplugs can cause symptoms similar to this also. Another thought would be one or several leaky fuel injectors or possibly bad injector seals. If the idle speed seems wrong, it could also be cracked/leaking idle air hoses under the intake runners. How much of the original 9previous) engine needed to be swapped over to the present motor?
Give the fact it's starting the same way, I too would tend to think of a chassis-mounted component though, like the accumulator.
Did you have the radiator cleaned and checked out to make sure this wasn't the source of the overheating?