They definitely can interfere with each other! For example,
Wouldn’t the magnetic field applied for the BEC-BCS crossover somehow cancel some of the magnetic field in the MOT and make it less effective (or complete ineffective)?
is completely correct- it would be hard, if not impossible, to investigate such physics with atoms in a MOT.
One very helpful way that people get around this is by trapping atoms in an optical trap with light that is far from any atomic transition, sometimes called an optical dipole trap (1). This kind of trap uses the interaction between the atomic electric dipole and light to create a conservative trapping force, so unlike a MOT it doesn't automatically provide any cooling. Instead, it is often the endpoint of an elaborate cooling procedure, which begins with a MOT. One of the main advantages of such far-detuned fields is that the force they apply to a given atom is generally insensitive to the internal state, such as the spin. Therefore, one can manipulate the spin or change the magnetic fields to do something like BEC-BCS physics without affecting the trapping force. Most optical lattices are similarly created with far-detuned lasers.
So using far-detuned lasers can make things much easier to understand, but other manipulation techniques can certainly affect each other. For example, applying RF to manipulate spin can affect a Feshbach resonance, as can near-resonant light. These have been used experimentally to modify the properties of resonances (2), (3). So these different tools can interfere with each other, but sometimes this can actually be turned around and used to extend their powers. Of course, sometimes this interference doesn't work out for what you want to do, in which case you might have to be careful about what fields you have on at the same time. As another possibility, sometimes two fields will affect each other in a relatively benign way, such as the shifts in energy levels from one field causing a shift in the absolute location of a resonance for another field, but not much else. So effects can range from nonexistent to extremely important to significant but only requiring a calibration.
A separate issue, which probably isn't quite what you had in mind, is the question of electrical cross-talk between all the different laboratory components. This is a practical matter rather than a fundamental one, but it can also cause huge problems. For example, RF radiation used to manipulate atoms might also be picked up on a laser circuit as noise, and cause some unwanted modulation of the laser. A great deal of experimental effort goes into isolating these systems as much as possible.
Here is a typical example of how it might all come together:
An experimental cycle begins by collecting atoms in a MOT. These atoms are then transferred to an optical trap consisting of intersecting far-detuned laser beams, and cooled further (with loss of atoms) by evaporative cooling. At the end of this cooling process there is a degenerate gas held in this optical trap. Then the optical lattice is turned on such that the atoms are loaded into the ground band. At this point the experiment itself begins, where the gas is probed (perhaps using RF or Raman transitions or any number of other tools), and then ultimately measured in a destructive way by taking an image using on-resonant light.