Sinking of the Bismark

Question 1 - Why not go on shooting? As long as Bismarck had not struck her colours she remained a legitimate target. There were many precedents and the Royal Navy was familiar with HMAS Sidney vs. SMS Emden at Cocos-Keeling and SMS Nurnburg vs. HMS Monmouth at Coronel. In both of these 1914 examples the winner shelled the crippled loser into submission or until it sank and in Monmouth's case, with no survivors. I would doubt that at the time, many gave leaving a second thought.

Question 2 - Ultra intercepts had indicated to Commander Home Fleet that at least one U-Boat was in the vicinity and more were on the way. It would have been criminally irresponsible to keep his ships on station to rescue Germans given that threat. The spurious "spotted periscope" story may have been an intelligence deception to protect Ultra although in fact U-556 (KL Herbert Wolfharth) actually spotted HMS Ark Royal but had no torpedoes left. The RN learned the hard way not to rescue survivors in U-Boat infested waters and it was the German's who taught them the lesson.

I suspect the post-war "Brotherhood of the Sea" mythology was a sop to the media. Nicholas Monserrat in "The Cruel Sea" probably gives a far better point of view of how the average RN personnel saw Hitler's navy at the time.