Would you please clarify if and under what circumstances one may kill insects or pests. On the one hand, it would seem that one should be able to do so. On the other hand, since these are G‑d’s creations, how can we justify taking life He’s given?
This is a delicate question, and it shows your sensitivity to the presence of G‑d in all levels of His Creation.
First, it’s important to understand that according to the Torah, G‑d created mankind at the pinnacle of Creation in order to give him mastery over all the creations under him. All of Creation was given over to him in order to fulfill man’s mandate as partner with G‑d. However, this dominion is not only a right but also a responsibility. We may only manipulate G‑d’s world, including its creations, according to His will and for the purpose of perfecting ourselves and coming closer to Him.
Since we can do this only while alive and healthy, we are allowed to take the life of anything that threatens life or health. Therefore, not only is one allowed to kill, for example, poisonous snakes or scorpions, it is actually a mitzva do so. This does not mean that one is allowed to go hunting them out of their natural habitat, because there they are not a threat. But when found among people, they must not just be left alone, but rather every effort (including killing them) must be made to prevent them from causing future harm. This is so even on Shabbat when ordinarily it’s strictly forbidden to kill anything.
Regarding things like bees, wasps, mosquitoes, ants or rodents, the answer depends on what extent they may harm us. In most cases, bees and wasps are not aggressive or particularly harmful. They also generally stay out of our way. It’s best to leave them alone when there’s no particular threat. Of course, people especially allergic to bee stings, or a wasp nest in reach of children, are examples of situations where one could definitely exterminate the threat. Mosquitoes, though less harmful, are very aggressive and bothersome. They can also get into food and cause it to be thrown away. Rodents may also bite people and waste food. Ants certainly disqualify food. Any of these may be repelled (preferable) or killed (if necessary). To avoid desensitizing us from taking life, it’s better to indirectly cause death than to actively kill them.
The above applies only in the case of threat to life or health, or significant discomfort or loss. However, to indiscriminately kill anything for no reason, or even to make no effort to avoid killing something, like stepping on an insect rather than over it, is a flagrant disrespect for G‑d’s Creation and an insensitive disregard for life. Even sitting on the lawn and inadvertently uprooting grass, or walking along and inadvertently tearing a leaf off a bush or tree shows great disregard for G‑d’s love of all creations.
Since this is true regarding even insects and plants, imagine how careful we must be regarding our treatment of fellow human beings!