When Loki pops back into the screening room, after re-acquiring the Tesseract, he lands on the floor, and drops the cube. From when he stands up, until after he dispatches Hunter B-15 back to the hallway, the Tesseract is not seen anywhere in the room. As he sits down on the side wall riser, it is back in his hand.
When Loki first arrives at the TVA, Cassie is seen at the front desk where the blue tesseract is placed on the counter. Cassie asks 'what is that', - however, in later episodes, when Loki roams the TVA, he runs into Cassie who claims that the Tesseract and the Infinity stones are used as paper weights. IE a common office supply item around the TVA.
Loki, as DB Cooper, is shown jumping from the plane during during daylight on a clear, sunny, day. In reality, Cooper jumped in darkness (approximately 8:13 PM in late November) and during a rainstorm. Also, Cooper removed his tie and left it on the plane as evidence, and made Schaffner and the rest of the crew stay in the front of the plane while he stayed in the back, unobserved, before jumping.
In real life, before DB Cooper jumped from the plane, he removed his clip-on tie and left it on the plane as evidence. When Loki jumps he is still wearing the black tie.
Loki tells Mobius that he is DB Cooper with a reenactment of Cooper's escape from a Boeing 727 in November 1971. However, the story is told incorrectly as the scene shows "Cooper" jumping from the plane during daylight, and his escape was actually at night in pitch darkness.
At 24'23", when speaking with Mobius in Theater 5, Loki breaks the 4th wall by looking very briefly directly at the camera.
After Loki is arrested, the paperwork clerk tells him that he has to sign a stack of papers, verifying that it contains his words. Then 2 more pages comes off the printer, and the clerk says that Loki needs to sign them as well. He signs the top paper, and none of the others.
The scene in Aix-en-Provence at around 12:30 supposedly takes place in 1549, but the stained glass window depicts Saint Joan of Arc. She's clearly identifiable by the attributes assigned to her (dressed in a white suite of armor combined with a skirt; carries a banner) and has a halo, a symbol reserved for saints. However, Joan of Arc was canonized a saint no earlier than 1920.