“Whether you eat or drink, or whatever you do, do it all for the glory of God.”
No, it is NOT inherently wrong to kill bugs, and the Bible doesn't say killing bugs is the sin of murder. The commandment against murder in Exodus 20:13 refers to unjustified, deliberate, and/or illegal homicide, which involves violence upon another human. That is not the same as killing insects.
In fact, in the Old Testament, the Mosaic Law specifies what insects are considered clean versus unclean, and the Israelites were permitted to eat the "clean" insects (Leviticus 11:20-23). Billions of people around the world today use insects as a part of their regular diet, and there is no sin in that.
Even if we're not hunting bugs for a savory snack, we do often kill insects for other reasons. Some struggle with the idea of killing anything while others will kill bugs indiscriminately. So what's OK and what isn't?
Killing Pests = OKAY
There are specific critters that are especially destructive and even dangerous to human life and/or dwellings. Some good examples are termites, bed bugs, ticks, lice, and poisonous spiders. It's 100% OK to kill these and other pests when they are a threat to the wellbeing of us and others we love.
Ridding our homes of these creatures can be a way of being good stewards of our living spaces and bodies, which are both gifts from God that we should take good care of.
While some people choose to get rid of these pests in more humane ways, such as putting a spider in a jar and taking it outside, we're not biblically obligated to go to this extra effort. What we should listen to is our personal convictions about any matter not specified in the Bible (Romans 14:22-23).
Killing to Watch Suffering = NOT OKAY
Remember that everything in this world is God's and that we are to be good stewards of His creation (Genesis 1:26). That means that needless killing and/or causing unnecessary suffering goes against our purpose to be good stewards.
For example, while wolf spiders might look scary, they're typically harmless to humans and act as predators for other actual pests like cockroaches. There's no reason to kill insects that pose no threat to you, your home, or your loved ones.
Killing God's creatures for entertainment or just to watch them suffer is wrong. Taking their lives for no good reason is a waste of potential for the purpose it was designed to serve—not to mention it's disrespectful and cruel treatment of God's creation (Psalm 145:9; Proverbs 12:10).
If we've been treating God's creation poorly or disrespecting Him as Creator of all, we can confess our actions to Him and trust that He will forgive us (1 John 1:9).
Motivations Matter
So, no, it's not a sin to kill insects that pose a threat to the health or safety or us or our loved ones. Killing insects by accident is pretty much unavoidable and should not be something we beat ourselves up about as it is not a sin issue either.
What the focus should be when it comes to bugs/spiders is our intent behind killing them. Everything we do as Christians should aim to glorify God (1 Corinthians 10:31), including how we treat His smallest and lowliest of creatures. Out of reverence for God as our Creator, we should treat His creation with intentionality and respect.