What is moulding?

Moulding is a strip of material with various cross sections used to cover transitions between surfaces or for decoration.
Is moulding stuccoed or painted?
Your choice. It can be painted or stuccoed but paint is prefered since it is easier to apply on the edges and corner of mouldings because of the detail. A darker or lighter tone then that used for the stucco is usually applied to give a nicer look.
Will mouldings crack?
No, not from temperature changes.
Are mouldings resistant to extreme temperatures and weather elements?
Yes, they are quite durable but quality depends on the manufacturer.