In the first scene where Kate Murphy is seen asleep, she is not wearing a bra. In the next scene where she wakes up she is.
The finger Kate uses to examine Sergei's scar jumps from pinkie to index.
The detective says the shirt Kate took was inadmissible. Since Kate was not acting in the capacity of a government agent but rather in the capacity of a girlfriend that was invited into the premises, the shirt would indeed be admissible. The worst that could happen is her being prosecuted for theft. He compounds the error by saying if she put it back he could get a warrant for it. Since the warrant would be based upon her "inadmissible" contact with the evidence, any such warrant would be just as inadmissible using his legally flawed logic.