Chakotay has variously described his "life's ambitions" as a historian, an archaeologist and in this episode a paleontologist. Paleontology is the study of the history of life on Earth and is based on fossils. It has nothing to do with the study of extraterrestrials. It's an Earth-based science.
At -260°C, all gasses except Helium would have liquefied inside the Ares, leaving some evidence floating around, yet there is nothing shown.
The photograph that Kelly looks at is still in the same shape when Seven examines it, after several hundred years of temperatures well below zero. It would not have been.
The "dark matter" asteroid is seen plummeting toward the anomaly. Dark matter objects are invisible.
In the opening scene, John Kelly seems to be referred by rank twice. The first time, they call him "lieutenant." A few sentences later, they call him "commander." Kelly was the mission commander, holding the rank of lieutenant. Both forms of address are accurate. Whether or not one would be used more than the other is another matter. The rank of Commander is derived from this.
When they first find the anomaly, Ensign Kim says that if it gets any closer, it will start ripping off hull plating. The ship is later shown to be much closer to the anomaly without any shield modifications or damage.
Chakotay mentions accomplishments in space including John Glenn, but omits any references to Russian cosmonauts such as Yuri Gagarin although he was the first human to actually fly in space, before Glenn or any other American.
In the opening teaser, John Kelly mentions that the Yankees have won Game 5 of the World Series, and then that one player has broken Joe DiMaggio's 56-game regular-season hitting record. While it is possible that things will change by 2032, at the time Voyager was filmed, events in postseason games (including the World Series) did not count toward regular-season records like DiMaggio's hitting streak.