A project I'm exploring requires placing a permanent magnet in close proximity to an RFID tag (within a cm or so). Can an RFID tag be read when there's a permanent EM field so close to it?
That really depends on your RFID technology and frequency range you are using.
With LF (125/135 kHz) and HF (13.56 MHz) you will have problems (as they work inductive) due to core losses, i.e., eddy currents (the higher the frequency, the higher the losses) and hysteresis.
You will have more problems with HF than LF. LF tags sometimes have ferrite cores to increase the inductance of the coil.
With UHF tags you may have even advantages, depending on the distance (frequency dependent) and orientation between tag and magnet and the antenna design of the tag. However, there exist UHF tags particularly for use on metal.
Depends on the tag. LF (125KHz) will certainly have problems. HF (13MHz) might work, depends on the tag's antenna. UHF (850-930MHz) tags will work, depending on their mounting with relation to the magnet. There are tags specifically designed to be mounted on metal surfaces, however I do not have much experience with those.