Does Shielding "Electrically" shield "Magnetically" too?
You would not be alone in this one. This is an often misunderstood phenomenon.
Static magnetic fields can not be shielded. They can be re-directed using ferrous materials but even those will not block them.
Electric fields on the other hand can be. Since an electric field is basically a voltage in space, they can not pass through a conductive plate that is held at a fixed potential. Space is shorted out as it were.
Alternating magnetic fields of sufficient frequency however, will not pass through a metal plate. The alternating field generates an eddy current in the plate which generates a cancelling magnetic field.
This is all explained in much better detail here.. Wikipedia
In a box, the distance from circuit-to-shield may not be adequate to develop an electro-magnetic-wave. In that case, you can validly consider the Efield separate from the Hfield.
The sea-of-mobile-electrons in metal is very effective for Efield shielding; the electrons roam to where needed on the metal's surface, to oppose the incoming Efield flux lines, coercing that flux to only impinge on the shield metal at exactly 90 degrees.
The ratio of Magnetic Permeability to Electric Permittivity hints at dramatically different effects between Hfield and Efield shielding.
Magnetic shielding varies with frequency. Standard 1 ounce/foot^2 copper foil of 35 micron thickness gives some attenuation (a few dB) at 5MHz. At 50 MHz, that same 35 micron provides sqrt(10) * dB/Neper, or 3.14 * 8.9dB = 28dB attenuation. At 500 MHz, that 35 micron provides 10.0 * dB/Nepers, or 89dB attenuation.
To begin to shield against 60Hz, you need sqrt(5,000,000/60) ~~ sqrt(100,000) = 316X more thickness; thus 35micron * 316, about 10,000 microns, or about 1 cm.
For magnetic fields, aluminum and copper have nearly the same behavior. Mu is same for both; differences appear from their different conductivity. Aluminum instantly tarnishes, so you cannot solder to it. Copper is easily soldered, using a big hot iron.
Regarding your question about noise-danger to nearby equipment, the answer is YES. Signals can interfere with each other. Check out my answer to "Distance between SPI traces....." question.
{edit} High voltage Efields cause lots of charge movement. If the frequency is low, you'll get detectable EXTERNAL movement of charges due to the Efield. In other words, the SkinEffect is your friend but SkinEffect only predicts attenuation; SkinEffect does not prevent external charge movement.