Are there connections between some properties in peoples faces and their behavior? I hear a lot of people making inferences about others based on their faces, but that's common sense. Are there really properties that could be deduced from ones face?
There is some research in the realm of sex differences and hormonal effects of aggression. The hypothesis meant by that is not so much the rather obvious one that male faces reliably cue "male" behaviour; although that is an important fact. For example, if a person has a beard, they are much more likely to be sexually attracted to women than to men. However, perhaps more in response to the intent than the phrasing of the question, researchers have speculated that "sexually dimorphic facial width-to-height ratio may be an 'honest signal' of propensity for aggressive behaviour". The idea behind this is that a common component, testosterone, underlies and causes both wide, masculine faces and aggressive behaviour.

I personally am not yet convinced of the reliability of this research - for example, in the progress of this research programme, more and more mediators are discovered, which by some might be interpreted to indicate an unreliable finding. However, the basic concept seems to be what you're looking for.

More generally, for now, even the linkage between hormones or neuroanatomy on one hand, and behaviour on the other hand is rarely clear, reliable and strong. Few of the variance in behaviour can be expected to be read off of facial configurations.