Props/set pieces in a side room off the ISS Shenzhou's transporter room have markings from Starfleet on them, not the Terran Empire.
When Burnham is in her quarters on the ISS Shenzhou and she is walking to her bed she had unzipped her uniform. Yet when the image cuts to her and Tyler having a conversation the uniform is fully zipped up then back down again. Clearly a mistake.
When Burnham removes her armor in her quarters aboard the ISS Shenzhou, it clinks as though it is made of metal but flexes like it's a much softer material.
"Captain" Tilly's mirror (hologram) stand-in's features are not flipped. In some shots where Tilly's face is not visible, the stand-in has the long and short sides of her hairdo on the wrong side to be a reflection of the proper Tilly.
Lorca orders Dr. Culber to transfer Stamets' care to another doctor. Lorca has no authority over medical matters. Only the ship's chief medical officer could make that decision, a character never identified onscreen.
Burnham laments that, if they are not able to return to their own universe and transmit the cloak-breaking algorithm to Starfleet, the Klingons will win the war. In Into the Forest I Go (2017), Admiral Terrel mentioned that Starfleet already had the algorithm and that it was being refined for deployment fleet-wide.