About as possible as it is to fly anything through the middle of a continuous nuclear explosion, though a controlled one. You'd have a bit of difficulty escaping the gravitational pull getting out and away from the other side, too !!
no. Are you just contemplating the fact that current technology has heat resistant material? Or hypothesising the sun not being solid? Here's another reason why - the sun is very far away and the crew would die of boredom before getting there