The jet is not coming out of the black hole its self but from the disk of matter around the black hole, magnetic fields focus the jet at the poles. The magnetic fields also originate from the disks because no magnetic forces coming from the black hole could escape the gravity.
The Quasar is caused by a supermassive black hole when the black hole has a vast amount of mater to feed on in the nucleus of the galaxy in which it resides, the act of this feeding creates the hot swirling disk around the back hole and these disks are where the massive amounts of energy come from. We call these galaxies in which they reside "active galaxies"
The Quasars them selves are the resulting energy sources we see coming form the active galaxies. That's the best definition I can give, they are kind of confusing to define the first time you begin to learn about them. Most definitions online say they are the nuclei of active galaxies, and that's true but its worded in a way that would be confusing to a lot of people. Aside from that a lot of people use the name Quasar to describe what it originates from, A very active supermassive black hole is often called a Quasar so your not wrong in saying its a supermassive black hole so long as your understanding that the black hole is causing the Quasar and that it is not the Quasar its self.
Also different types of Quasars depend on how your viewing them, for instance you would see a Blazar when one of the jets was pointed directly toward you.
All most every quasar we have observed has had a host galaxy in which it resides, in fact only very recently have we seem to have found one with out a host galaxy.
http://www.astronomynow.com/news/n0911/30bh/ As far as I know this is the only quasar ever to be observed outside of a host galaxy. So every other quasar except this one has been observed to lay within a host galaxy.
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