What is the effect of heavy loads on the electrical grid?
Firstly, common types of motors take a lot of current when they're stationary, as they are when you first turn them on.  This isn't really 'because of the coils' particularly - in fact, a pure inductance, which is what one tends to think of as a coil has exactly the opposite behaviour.  It's because they're frantically drawing current to try to get themselves up to speed.  

So, your neighbours draw a lot of current when they turn on a motor, and because you and they clearly share a cable back to some more major part of the mains system, this high current creates a voltage drop across that cable which you can see on your lights.

You're right to think that a similar effect happens at all levels of the supply network, but two things lessen the visible effect:


Even big industrial loads tend to be a small proportion of the total load averaged over lots of properties
The supply for a big area has an incredibly low impedance, so that normal loads don't cause much effect on voltage.


Electricity suppliers do have other tricks to regulate the voltage they supply - they change tappings on transformers, and they manipulate the flow of reactive power through the system, both of which affect the voltage at their customers.  But mostly the reason you don't see flicker at the level of a whole town is just that the supply is very stiff relative to the individual loads.

Really, really massive single point loads do have to liaise with the network before switching, but those don't tend to be at the level of a few motors in factories.