Tempo SC Ultra is considered a general use insecticide and is labeled for outside and inside use. It is odorless, and very safe to use around children or pets. The general rule is that while spraying you keep children or pets in another room or outside to play. After about one hour it should be dry and it is perfectly safe to allow kids and/or pets back in the room. This product is only strong enough to kill tiny insects, and would have no effect on animals when used correctly.
Yes, Tempo SC Ultra is labeled to kill spiders. Spiders are very difficult to control and most residual insecticides like Tempo SC Ultra will kill any spider that it is physically sprayed on, but new spiders entering the area are very difficult for any insecticide to control. This is because spiders bodies are high off of the ground and they do not groom their legs like other insects do. How these products mainly control spiders is that they kill the spiders prey (other insects), and if there is no food around, then the spiders will either starve to death or move on.