When did Crusher develop such an interest in shield technology? That falls under the aegis of the physical sciences, and she has demonstrated an almost aggressive disinterest in such things whenever Wesley or Picard has tried to discuss them with her.
As the shuttle equipped with the metaphasic shielding flies into the star's corona, the temperature rises to more than one million Kelvins. This may be incorrectly regarded as an error because the surface of a sun-like star is only around 6000 Kelvins. However, the corona, which is an aura of plasma that stretches out thousands of miles from the surface, is in fact much hotter than the surface. Our Sun's corona is well over 1,000,000 Kelvins in temperature.
Dr. Crusher says the reason she's not allowed to do an autopsy is that "Ferengi traditions don't allow violation of the integrity of the body before burial." Later in Deep Space 9, we find that a Ferengi tradition is to have their body freeze-dried, ground up, divided and sold. However, that particular rite of burial is alluded to only being practiced by high-ranking Ferengi, not an overlooked science worker. There would be no (financial) interest in his remains.
Because the security of the ship might be at stake due to a potential murder, Picard should have allowed an autopsy on this matter. However, Picard himself states to Beverly that her theory about a murder isn't enough evidence to support overruling the Ferengi customs.
At one point, Data refers to the star as "the sun". It is done ever so often on this show; nevertheless, it is technically a "star". Data of all people should use the correct technical terms, especially when on duty on the bridge.
However, he referred to it as "the sun", not "the Sun". The star is *a* sun and is *the* sun of this system.
Near the end of the show, just before Deanna tells Picard that she is concerned about Beverly, the reflection of the boom mic can be seen passing in front of a panel near the Ready Room, in the top right-hand side of the screen.