Question: When Kirk and McCoy try to rescue Chekov at Mercy Hospital, Kirk removes the 20th Century medical team into an adjacent room and uses his phaser to instantly fuse the metal door lock. The medical team cannot directly see Kirk do this, as they are visibly several feet away on the other side of the door. It's also safe to say that the medical team has never seen a phaser and can't comprehend its function or capabilities. As Kirk turns away from the door to rejoin McCoy, the trapped medical team only then rushes up to the door, and the trauma surgeon exclaims, "He melted the lock!" However, it seems that you'd have to laboriously dismantle the doorknob to determine that the lock's internal components were fused. So, how did a 20th Century surgeon deduce at a glance that Kirk had somehow melted the lock?
Answer: The doctors were watching through the window the entire time. There was a visible red laser beam from the phaser, culminating with a puff of smoke or vapor emanating from the knob. It wouldn't be a huge leap for anyone to surmise that the knob had likely been melted.
Try watching the scene. No doctors are looking through the window when Kirk phasers the door lock.
Answer: Or perhaps the part of lock on the doctors' side is visibility melted.
Answer: The knob would have been super-heated by the phaser blast. Enough that it could be felt without touching, and he simply could have come to the conclusion that a metal object that hot would likely have its internal components melted without a systematic analysis of the doorknob. He's also a surgeon and needs his hands. He wouldn't last long at the job if he was someone who went around putting his hand on glowing-hot doorknobs.
He could've also been guessing as it appears he tries opening the door. Why they don't break the glass is beyond me, but that's a character mistake, and not up for debate here.
Question: Why does the probe even wish to talk to humpback whales?
Answer: It's stated by Spock in the movie, even if he is just speculating. He mentions that humans are not the only intelligent special on Earth, and that it's human arrogance to assume that the probe's signal, "Must be meant for man." The point is, the aliens were communicating with the whales, and when that communication stopped, they sent the probe to find out why.
Answer: There is a hypothesis by Spock that the probe was perhaps sent to find out why they didn't hear the whale song any more.
Answer: Wouldn't you if you could?
Answer: The lock, and the area around it, would have become hot as a result of melting the lock. The hospital staff would then jump to the conclusion that the lock was melted. The real reason they mention it, however, is so the audience knows what he did to the lock.
But you would think, if the doorknob was still searing hot two seconds after being fused, that the first thing out of the surgeon's mouth would be a scream of pain, rather than "He melted the lock!"
Charles Austin Miller