Downgrading from a dimmer to a switch
If the switch has screw terminals as opposed to wire leads you DO have to pig-tail a piece of wire to the splice and then connect that tail to the switch. DO NOT place more than one wire under a binding screw terminal, unless of course it is designed for it. Even in that case it would only be designed for up to two wires.

Do you have a model number for this switch?

Regarding the neutral wires in the box, NO not every wire goes back to the breaker box. This is why you have several whites. One is a feed and the others are feed out to other items on the same circuit. If there is more than one circuit in the box then the neutrals from those different circuit should NOT be spliced. The only time this is not the case is if it is a multi-wire branch circuit (shared neutral).