It depends on the "quality" of the Faraday cage, and whether the magnetic field is constant or varying. The ideal Faraday cage is a superconductor, and it blocks external magnetic fields by creating electrical currents on its outer surface which redirect the magnetic fields so they basically don't penetrate. Normal conductors aren't so good at blocking magnetic fields by the same technique, because normal conductors are electrically lossy: they can't maintain current flow. Instead, to block magnetic fields, people often use a material similar to mu-metal, which is a specially annealed cobalt-iron alloy which is very good at "collecting" magnetic field lines. The magnetic fields are concentrated within the mu-metal; the field lines are along the mu-metal rather than penetrating through it as they would most other materials. At high frequencies, like used in microwaves or cell phones, normal conductors work well for a Faraday cage. The fluctuating magnetic field generates a temporary electric field in the conductor, and the flow of electricity creates an opposing magnetic field. This is the same principle as in the case of superconductors mentioned above, but the current is short-lived so lossy conductors are okay.

Good question. I had to do a bit of research on the topic because it is not my field of expertise. But I did find the following Wikipedia excerpt that speaks directly to your question:"To a large degree, Faraday cages also shield the interior from external electromagnetic radiation if the conductor is thick enough and any holes are significantly smaller than the radiation's wavelength. For example, certain computer forensic test procedures of electronic components or systems that require an environment devoid of electromagnetic interference may be conducted within a so-called screen room. These screen rooms are essentially labs or work areas that are completely enclosed by one or more layers of fine metal mesh or perforated sheet metal. The metal layers are connected to earth ground to dissipate any electric currents generated from the external electromagnetic fields, and thus block a large amount of the electromagnetic interference."

That's from
Faraday_cage
where there's a lot more useful information about Faraday cages.

Good question - I'm not entirely sure, although I would guess the answer to be "no", since the reason why the cage blocks radiation is that the radiation stimulates an electric current in the cage itself, thereby dumping its energy into the cage and not what is inside of it. I might think that if the cage were not only composed of an electrically conductive material, but a magnetic material as well, then it might affect magnetic fields, but that might amplify them rather than absorb them.

Here is an experiment that you could do to find out: get yourself a large, thin sheet of a magnetic metal (e.g. steel). Attach a magnet to one side of the sheet. Put a movable piece of magnetic metal on the other side of the sheet. If the movable piece of metal is attracted to the magnet, then the magnetic field is penetrating the sheet, and would penetrate a grid like a Faraday cage as well. If the movable piece is unaffected by the position of the magnet, then that means that the magnetic field cannot penetrate the sheet.