Yes, you thaw out the mouse completely before feeding it. I like to let my mice/rats thaw on a heat mat for a couple hours in the snake room. By the time they are completely thawed, all the snakes are ready to eat.

I usually just dangle the mouse by the tail and they snatch it up rather fast although sometimes I will have to set the mouse down in the enclosure and leave the room before the snake will eat it. I usually dangle for 5 minutes and if they haven't struck, I leave the mouse in there overnight. If I leave it in there overnight I throw it out in the morning.

I wouldn't bother refreezing the hoppers since your snake will quickly outgrow them.