Jeffrey Archer's "False Impression":
1, referring to FBI's training in Quantico:
His class supervisor at Quantico would carry out a surveillance detection run every evening after class, when he would follow one of the new recruits home. If you managed to lose him, you were singled out for a commendation. Jack went one better. Having lost him, he then carried out an SDR on his supervisor and followed him home without being spotted.
question: what does "lose" mean here? SDR? go one better?
2. Anton held up his hands in mock defeat, grinning widely.
question: in mock defeat?
3. (in Bucharest) Anton once again took her by the hand and led her back down the corridor towards the staff refectory. When they entered the senior common room, Anna was greeted by the sound of good-humoured chatter as tutors swapped anecdotes while they sat around in groups enjoying nothing stronger than a coffee. They didn't seem to notice that the furniture, the cups, saucers and probably even the cookies would have been rejected by any self-respecting hobo visiting a Salvation Army hostel in the Bronx.
question: I don't understand the last sentence
thank you for your help.