Having the output as headphones how are the passive pickups powered in an effects processor ? Do they take up the power source of the processor itself ?
The pickups are "powered" by the movement of the strings. The effects processor amplifies the signal until it's powerful enough to be heard. If you plugged your headphones in directly, there wouldn't be enough signal even when you bang hard on the strings to hear anything through your headphones.
A low level signal is generated by the vibrations of the strings above magnets. This signal is then boosted by a preamp at some stage in the process, whether it is an effects pedal (in your case it sounds like it is), or a power amplifier.
Active pickups differ in that they have a built in, powered preamp which gives a higher level signal before it gets to an effects chain or amplifier. If your guitar has a battery in it, it almost certainly has at least one active pickup on board.
It may be useful to view magnets as pushing out a stream of "north-pole stuff" out the north pole which wants to find a path to the south pole, and "south-pole stuff" which wants to find a path to the north pole. In practice, there's only one kind of flux and it doesn't really "travel" from one pole to the other but rather connects them, but imagining flux as "flowing stuff" may be easier than viewing it more abstractly.
Flux repels itself, which means that while a tiny bit of flux may be able to travel from the north pole to the south pole right against the magnet, much of it will have to travel some distance away from the axis of the magnet before traveling from one pole to the other, and then travel back to the magnet. Because flux travels more easily through steel than through air, the string will allow some flux to quickly flow from a point very near the magnet's axis to a point some distance away, thus making it easier for the flux to get away from the magnet.
Each unit of magnetic flux which encircles a wire as it connects the north and south moles of a magnet will induce a unit of electric flux in the wire; changes in flux translate into voltage. If a wire is looped into a coil, flux which encircles all the wires on one side of the loop but not the other will induce flux (and thus voltage) on all those wires, and all the induced voltages will add together; any flux which encircles wires all the way will induce flux at different parts of the coil, but some of those parts will cancel others, having no net effect.
A single-coil pickup works by having the spring spread out some of the flux so that it goes through the magnet (which is inside the coil) but returns some distance away (outside the coil). The closer the string, the greater this effect.
Humbuckers have a more concentrated effect since it's possible for flux to connect the north pole of one magnet to the south pole of another, and the north pole of the second magnet to the south pole of the first. In a humbucker pickup, north and south poles of opposing magnets sit next to each other in opposite coils. Thus, any change in the flux which flows through the magnets will affect the flux flowing through the two coils in opposite fashion. If the coils are wired opposite, this will cause any generated voltage to to be added together.