What's Past is Prologue - S1-E13
Question: Why are Joann Owosekun and Rhys still wearing the Terran uniform while everyone else on the ship has switched to the Federation's as they were supposed to?
The War Without, the War Within - S1-E14
Question: In dialogue, Starbase 1 is described as "100 AU from Earth," which is over three times the orbital distance of Neptune. But the onscreen graphic shows the starbase next to a planet that can be visually recognized as Earth; portions of North America, including the Florida peninsula and the Gulf of St. Lawrence, are visible. The distance given in dialogue makes sense, given the episode's plot; it's a distant location still within the Solar System, and the Klingon attack is said to be "practically in Earth's backyard." The location shown in the graphic does not make sense, because Earth would already be under attack yet everyone is concerned only about the starbase, not the planet (which is never mentioned at all, let alone said to be Earth).
Answer: The planet shown could well be a terraformed moon on the outskirts of the Solar System. The planet is not shown in great detail behind the damaged station.