A fast idle of 1000 rpm when the engine is cold (like the first time start in the morning) is quite normal. Your car is trying to heat up the catalytic converters to reduce emissions. As long as it goes down to normal rpm, it is fine.
The coolant sensor sometimes get covered by some gunk over time, which slows down the response to coolant temperature increase, so the warm up time as perceived by the engine computer increases. If it has not been changed for a while, it would be something your mechanic can remove it and inspect it, both physically and electrically (resistance measurements with the sensor dunked into water at varying temperatures). A replacement is recommended as they are not expensive. Also your mechanic should check the intake air temperature sensor readings (the sensor is in the MAF unit). The readings are available through the OBD2 port to a engine scanner.
You can ask the mechanic to clear the codes (even if there is none) for the engine computer(ECU) with a scanner. This will reset the ECU, and the ECU will go through re-adaptation and re-learn idle speed and many other parameters as you drive. You can do the same by removing the negative terminal of the battery for 10 minutes, and with key out, reconnect the battery, and insert the key and turn it to the last position before the engine cranks (run" position). Wait 3 minutes, turn back and remove the key, insert the key and start the car. That's it. With this method however you need to know your radio code prior, AND know how to re-synchronize the windows, sun-roof etc.
Probably after the battery died, and replaced, your engine computer went through adaptation and re-learn.
