I'll try to catch and release spiders, crickets, moths, and mosquito hawks, if it seems like it's worth the effort. In most cases in the house I'll just leave them be. At the dance studio or out in public I will try to catch and release the bugs that seem to be freaking other people out.
I have a funny little ecosystem going on in my house... there are spider webs in some corners, and the spiders eat some of the little bugs, and then the geckos eat some of the spiders, and then the dogs eat some of the geckos. Of course, the dogs also eat some of the spiders and little bugs if they can catch them!!
Ants get no quarter from me, but then I live in fire ant country. I spray to keep them out, and set bait traps when I do see them in the house. Palmetto bugs, mosquitos, and cicadas also get no quarter, and I will kill them if they get in.
My dogs are big on bringing in critters from outside (we have a dog door). They bring in things like june bugs, cicadas, toads, even a snake. Usually they'll kill and eat the bugs, and play with the bigger critters till I figure out what they're doing and take it away from them.
I love my dogs. Really...