How can I get alerts when the ecobee turns on the A.C. along with the current temp and time when it does?
CoRE can probably do this, monitoring thermostatOperatingState changes to ‘cooling’ and the temperature, but because the ecobee API is polled, the alert will be delayed. The length of the delay will depend upon the DTH you use for the ecobee and its polling frequency.
With more work, you could write a SmartApp that queried the history from the ecobee API to extract the actual start/stop times and temperatures. This is what the Ecobee App itself does to show the thermostat history.