Is the bleed air passengers breathe tapped from the engines before or after it comes in contact with fuel? And why? Isn't this unhealthy?
The bleed will be tapped from one or two of the 3rd, 4th, 5th or 6th High Pressure Compressor stages.  Usually there are two bleed ports.  The highest pressure port will supply really hot air for things like anti-icing.  The lower pressure port farther upstream will supply air for air conditioning/pressurization and other pneumatics.  Some systems combine the two ports into one system and draw from the low pressure or high pressure one as required.

The air has to come from the compressor, not after combustion, mainly because it needs to be clean ambient air free of kerosene combustion byproducts, and it's plenty hot enough anyway and taking it from the turbine would be way too hot.  The discharge in the last stages of the HP compressor can be upwards of 700-800F at takeoff thrust.  

If the aircraft uses a high temperature "evaporative" anti-icing system, it will tap air from hotter stages of the compressor than if there is a non-evaporative system (Evaporative = anti-iced skin temperatures above the boiling point of water.  The leading edge will be over 100 C, like the surface of a steam iron.)

Even if it is taking "clean" air, the bleed flow will have all sorts of charred bits of things from the atmosphere in it; pollen, bits of bugs, that collect and get everything dirty in the ducting and components (at lower altitudes).  Combusted air would be ten times worse, not to mention the carbon monoxide poisoning the passengers and being way too hot to be usable in the first place.