Is it prescribed to say "Bismillah" (in the name of Allah) before killing harmful insects and animals, whether they are "fawasiq" (transgressing/harmful creatures) or others, or should they be killed directly without saying "Bismillah"?

The Prophet, peace and blessings be upon him, did not command uttering the Basmalah (Bismillah) when killing noxious animals. Had it been prescribed, he would have clarified it, just as there is no independent evidence for it.

The warning (tahrij) is mentioned only for snakes found inside homes, and it is specific to them. It means restricting them by saying, "I adjure you by Allah and the Last Day not to appear to us, nor to harm us," three times. If it does not leave after this warning, it is then killed without uttering the Basmalah. As for snakes outside homes, they are killed immediately without a warning.

The Basmalah is prescribed when slaughtering for food, not when killing harmful creatures.