Sarek mentions to Burnham that he has spoken to the High Command, but the High Command was disbanded over 100 years previously.
The crew of Discovery is able to identify the Hiawatha by having Saru, who is said to have better visual acuity than humans, read its registry number off the viewscreen. The image on the viewscreen was an enlargement of a digital image. Saru would not be able to pull any more visual information out of it than exists in the computer.
That was a lot of trouble to go through to obtain some rock with dark matter inside it, when all ordinary matter has vast quantities of dark matter flowing through it all the time without being noticed, since it's completely intangible.
This episode introduces "turbolift space" into the Discovery, a massive chasm through which the turbolifts are seen traveling that is far too large to fit anywhere within the ship. All other Star Trek series show turbolifts traveling through shafts in the walls of the ships, akin to elevators.
In the turbolift, Michael asks Linus how they are doing, to which they reply in an untranslated language. Michael responds "Yes, I hear that's going around." Since the turbolift is within the Discovery, and Linus is part of the federation, the Universal Translator should have translated the conversation for us.
The distance Burnham, Pike, Nahn, and Connelly travel inside the pod launch tube is far greater than the distance from the entrance of the launch tube to the rear of the ship.
Reno says she identified Pike, Nahn, and Burnham by their "Federation insignia." They are wearing Starfleet insignia, not Federation. Starfleet is an exploratory and defense agency, the Federation is a governing body.