Right after Burnham has finished talking to Sarek we see the USS Shenzhou fly past undamaged. In the previous exterior pull away shot there was major damage and large and sections of the hull were missing around the brig.
The crew of the Shenzhou beams a photon torpedo warhead onto a Klingon casualty so that it will be taken into the Klingon ship via tractor beam as the Klingons gather their dead. Photon torpedo warheads contain anti-matter, which Star Trek lore has established cannot be moved via the transporter. It must be physically moved in special containment units. Attempting to transport the warhead would theoretically beam away the components of the warhead that are comprised of normal matter, exposing the anti-matter core and detonating a large section of the ship.
Burnham is shown here transferring to the Shenzhou as a favor to Ambassador Sarek, rather than attending Starfleet Academy, A non-commissioned officer could not attain the rank of Commander. They have a separate ranking system of petty and warrant officers.
When Admiral Anderson's holographic image is first seen in the Klingon ship, the perspective on his image does not match the angle of the shot, making him appear as a flat image floating in the room while appearing dimensional in all surrounding shots.
Burnham and Georgiou lament how Burnham's actions led to conflict, but Georgiou stopped Burnham and confined her to the brig for mutiny before she could carry out her planned attack.
Only Burnham and Georgiou beam over to the Klingon ship in an attempt to take T'Kuvma prisoner. Not only is it against protocol to send the top of the chain of command on such a dangerous away mission, but they sent only the two of them onto a ship full of armed and angry Klingon warriors.