Thermal Equilibrium is when two objects of the same temperature are touching, and no heat is transfered between them.
Yes. Thermal Conduction is a solid's ability to transfer heat. Like when you have a metal spoon and a plastic spoon, the metal spoon feels colder because it's better at drawing heat away from your body.
This is why everyone tells me to put a cold spoon in my hot cocoa when it's too hot. The metal spoon cools down the drink by taking some of the heat away!