Reaction force from magnetic force A charged particle moving in a magnetic field experiences a force which will change its momentum. A search on the Internet gives the explanation that the field itself carries the lost momentum. Will the field then exert some force on the object creating the magnetic field (e.g. the magnet/current-carrying wire)? If not, can I say that by attaching a magnet to a coil carrying current, the whole set-up can theoretically lift itself up?
 A: 
Will the field then exert some force on the object creating the magnetic field (e.g. the magnet/current-carrying wire)? 

Maybe, maybe not. The field momentum might end up being taken in by the wire, but it can also be radiated away. 

If not, can I say that by attaching a magnet to a coil carrying current, the whole set-up can theoretically lift itself up?

At least in principle, yes; this is how solar sails work. As a slightly more realistic example, a spacecraft can turn on a laser (whose light carries momentum) and thereby be propelled in the opposite direction.
A: Yes, of course the magnet will experience an equal and opposite force to that on the charged particle, by Newton's Third Law.
This is intuitively understood by considering the example of a typical motor that uses permanent magnets. Of course, the 'stator' will experience an equal and opposite torque to the 'rotor' - the stator needs to be 'held' or otherwise supported to prevent it from rotating. It is also quite possible to hold the 'rotor' stationary instead and observe the 'stator' rotating - think about what happens when an electric drill jams while you are drilling into something.
No, electric motors are not exempt from Newton's Third Law! ;-)
