Is it possible to separate the poles of a magnet?

Well, in order for this splitting to be possible, the magnet would have to be made of two magnetic monopoles (like charged particles, but with "magnetic charge" instead of electric charge) bound together. No known magnet is actually constructed this way; all real magnets that have been studied are made of either little current loops, or particles that have a spin magnetic moment (and these basically act like little current loops).

It's still an open question whether or not magnetic monopoles exist. Some theories predict that they should, but most have nothing to say about it either way. I am not aware of any theories that prohibit the existence of these monopoles. Quantum field theory in general falls in the second category; that is, there is nothing inherent in QFT that requires magnetic monopoles to exist or not exist.

This is basically how a magnet's atoms look like:

so, when you split it into two, you do not change anything but the length of the magnet. As you can see the North poles(Black sides or "K" as it's in a different language) face north and south poles face south in each part of the magnet.

IF there are such things as monopoles, then it is possible to define a magnetic charge which would allow us to separate a unit magnet into two monopoles. However there are no confirmations for magnetic monopoles so we'll have to accept magnets as a whole for now.

As for black holes, their attraction is gravitational, and not very different from the gravity earth applies on you.

Magnetic monopoles certainly exist. This does not require a GUT, they exist in any theory where the electromagnetic U(1) is compact (i.e. where charge is quantized). This follows only from the semiclassical behavior of black hole decay, and so does not require unknown physics.

The reason is essentially the one you state--- you can polarize a black hole in a strong magnetic field, and let it split by Hawking radiation into two oppositely magnetically charged black holes of opposite polarities. Magnetically charged black holes exist in classical General Relativity, as are arbitrary electric-magnetic charge ratio holes, and you can't forbid them, at least not for macroscopically sized black holes, without ruining the theory.

When you let the monopolar black holes decay, you find relatively light monopoles. The lightest monopoles will be lighter than its magnetic charge, so that two such monopoles will repel magnetically, not attract. Presumably, the monopole you find will be a (small multiple of) Dirac's magnetic monopole quantum.

To me, this is as certain as the existence of the Higgs. We haven't observed either one, but the theoretical argument is completely convincing.