Will I get more pests/insects if my neighbor sprays and I don't?
Pests (and all living things I suppose) tend to populate their environment to capacity.  If any action drives them to seek new territory, and that territory is already at capacity, then any population spike will only be temporary.  Population dynamics is a complex topic and no real-world system is inherently stable or predictable, so take this with a grain of salt.  But I'm pretty sure the pest control company is just giving you a sales pitch.

As for "orgainic" sprays, you'll have to be specific about the type of bug you are spraying for.