Book and Burnham transport out of the merchant lying down but rematerialize standing up.
Human hair grows at an average rate of about a half an inch per month, or about 6 inches per year. Burnham's hair is shown growing exponentially faster than that, reaching a length of several feet in just 12 months.
Book tells Michael that she damaged his dilithium recrystallizer and so he needs more dilithium to get underway. He then explains how the courier system works: the Orions pay each courier just enough dilithium to complete their job. Recrystalizing dilithium allows the same portion of dilithium to be used indefinitely. If Book has a dilithium recrystallizer, the technology must be widespread, and thus each courier could continue to operate without the constant need to scrounge for more dilithium.
It is shown here that the Red Angel suit can function perfectly well based on pre-programmed instructions, making Burnham and Discovery's sacrificial trip through time fairly meaningless.
Aditya Sahil tells Burnham that only a Starfleet officer is allowed to raise the flag while presenting her with a Federation flag. Any Federation citizen may raise a Federation flag. The Federation and Starfleet are not the same thing. The Federation is a governing body and union of citizens, Starfleet is an exploratory and defense force.
Aditya Sahil tells Burnham he was never sworn in as a Starfleet officer because there has been no one to do it his whole life. His scans showed several Starfleet ships in the sector. If Burnham has the authority to commission him, so would any of the officers aboard those ships.
Antique dealers in the Star Trek universe would have scanners to help verify the authenticity of the items they buy and sell. Such scans would show that Michael's "vintage" Starfleet equipment was, at most, only a few years old, as she brought it forward in time with her, not over 900 years old, as an antique dealer would expect it to be.
Because Discovery did not arrive at the same time as her, Burnham and Aditya Sahil reason that, due to the random nature of time travel, it may arrive the next day or in a thousand years. Neither brings up the other obvious possibility: that it arrived well before her, possibly so long before that all of her crewmates are dead, possibly even before the Burn and it was destroyed along with most of Starfleet.
Michael states that dilithium is "the heart of every warp-capable ship." That is not true. While many warp drives do indeed rely on dilithium, there are some known to run on other power sources. Romulan ships, for instance, use an artificial singularity as their power source.
Book mispronounces the word "temporal."
Michael is in disbelief that the Federation has dissolved, stating that such a thing is "impossible." She has moved nearly a thousand years into the future. It is incredibly rare for a governing body to last anywhere near that long.
Book tells Michael to cover her "uniform" to blend in. She is not wearing a uniform. She is wearing a unique under-suit that was part of the Red Angel's function. The only thing that denotes her as a member of a service branch is her Starfleet badge, which she does not remove and is visible beneath the coat Book gives her to wear.