Is it obligatory to say "Bismillah" for each shot when hunting with a rifle that stores five rounds, or is it sufficient to say "Bismillah" at the beginning of the hunt, especially when there is a flock of birds? In such a case, should "Bismillah" be said for the flock or for each shot?

You must say "Bismillah" (in the name of Allah) with each shot, because the hunter is commanded to do so when sending (the hunting animal) or shooting, just as the slaughterer says "Bismillah" when slaughtering. This is based on the hadith: "If you send your dog and say 'Bismillah,' then eat (what it catches)." This "Bismillah" is obligatory according to the more preponderant opinion, provided one remembers and is able. There is no harm in adding "Allahu Akbar" (Allah is the greatest).