When is the autopilot engaged and disengaged?
I am a B777 pilot. On departure, we generally hand fly until about 10,000' to keep up our skills. If the departure procedure is complicated, we will ask for the AP on at 200' so there is no chance of messing up the departure procedure. 

On approach I typically disconnect the Autopilot at about 1500' AGL once visual contact is assured and I am inside the Final Approach fix. 

If there is any chance of a possible missed approach due to weather, or other conflicting aircraft, on the runway or elsewhere, I will leave the AP on until there is virtually no chance of a missed approach.   

Most of the pilots I fly with tend to do the same.