If a spider or another bug can bite someone or get into food or frighten children. What can be done? Can it be swept away or trapped with a cup or grabbed with a tissue and thrown away?
The spider or bug can be swept away.
Regarding trapping, it can only be trapped if it is really scares the children! There is no heter to allow trapping in order to prevent the spider from getting into food.
It cannot be picked up since it is Muktza.
All the best
Swept away- Mishna Berura 308:115 (sweeping is considered tiltul min hatzad), Shulchan Aruch 311:8 (tiltul min hatzad is allowed for the benefit of a non-muktza item)
Trapping- Mishna Berura 316:27, 316:37.