In Season 3, the Discovery meets the Andorian character Ryn whose antennae were cut off by Osyraa, leader of the Emerald Chain, after he once tried to mount a rebellion against her. However, Andorian antennae grow back in a few months after being cut off, as stated by Shran in United (2005), who had one of his cut off by Captain Archer. However, Ryn's antennae never grow back in his various appearances throughout the series. However, given that 1000+ years have passed between the two incidents, it's likely that a method of mutilating the antenna to ensure they do not grow back was created for ultimate humiliation to the Andorians.
When Hugh asks Paul why he reached out to Adira, he states that the main reason he did so was because Adira also loves someone who died but isn't really gone, a situation that Paul experienced with Hugh. Paul didn't find out that Adira could still see and interact with Gray until the conversation they had after Paul made an effort to reach out to them. However, it is well known that Trills have access to all the memories and experiences of the previous hosts via the symbiote. Essentially, they are very much a part of Adira and therefore are not really gone. Adira telling Paul that they still see Gray is irrelevant.
When Burnham and Book enter the turbolift to go see Saru, neither of them gives a destination yet the lift starts moving anyway.
Book programmed his ship to take Grudge to Burnham on the Discovery if he didn't return within 24 hours. According to dialogue, the ship took 3 weeks to get to Burnham, yet when Burnham and Georgiou take that same ship to go find Book, they are able to arrive at their destination, find and rescue Book, and return to Discovery in a matter of hours.
Burnham says that she "reverse-engineered" the system that Book uses to find Grudge if she ever gets lost, so that it will instead find Book. Reverse engineering means duplicating a product or process by examining the finished product and working out how it was accomplished or made. What she did was reverse the process (used Book's locator as a beacon), not reverse-engineer it.
Paul refers to Adira as a "tween." A tween is an adolescent that has not yet reached their teenage years, usually defined as between 10-12. Adira is, as this same scene iterates, 16, and thus by definition is not a tween.