It's all down to who makes the lock, and what level of security. No-name locks generally don't have security pins in them, but I've come across the odd one here and there. There isn't really an average amount, although you will get an idea of the type of pins and how many there should be in specific types of lock after you pick a whole bunch of different ones.
Spools are usually easy to spot and when the lock drops into false set it'll be pretty obvious. It's not always visible, but you'll definately feel it. It can be trickier depending on the pin though... a spool with a really shallow cut, for example, will feel like a properly set pin because the plug will click and rotate just a tiny amount like with a standard pin. This is what I would call a "true" false set.
In the case of obvious false sets, one of the pins will cause the plug to try and rotate back in the opposite direction, just like LSA said. This isn't necessarily the pin you just pushed on to make it go into false set, so feel around until you feel that plug trying to fight its way back the other way.
This counter-rotation happens nomatter what, but if the false set isn't really in your face then you won't feel it as much. Regardless, all you need to do is push down on the false setting pin and counter that backwards rotation with your tensioner until you feel another click.
As for the more subtle ones, if you set a pin and nothing else is binding, go back to that pin and push it until it clicks again. Sometimes you'll need to repeat this more than once for the same pin, in the case of serrated drivers.
Here you can see a lock with the plug at its rest position.
And applying tension - there will be a slight give, but no dramatic rotation.
Next you will see an obvious false set. This is an indication that the lock is almost fully picked, although it'll do this early on if the key pins are all of similar height because the drivers will all be half way across the shear line. They will have a binding order just like normal pins; the only difference is the feedback.
And of course the gratuitous shot of the lock in its fully picked position