You may be able to temporarily create a system like this, but long term, it would not be stable.
Let's first reduce this system to a binary system to talk about what would happen.
In a binary system, a planet has one of three orbit options:
- It orbits both stars in a figure-8 pattern, passing through the L1 point.
- It orbits both stars in a giant elliptical orbit. (P-type orbit, or circumbinary orbit)
- It orbits one star in a tight elliptical orbit. (S-type orbit)
You can see in the potential diagram above how the figure-8 pattern is stable, but any perturbation would cause it to change into one of the other two orbits.
This problem is exacerbated in a 3/4/more star system. And if you have more than one planet, then you're guaranteed to get the kinds of perturbation that will decay all orbits into S- or P-types.
However, we do know about binary star systems with multiple planets (Kepler-47 is a great example), so this is not an impossible feat. Be warned that more than a couple planets may be impossible, unless they are significantly far from their parent stars.
As far as habitability goes, we honestly don't have a full understanding of the Goldilocks zone around multiple star systems. If the two stars are far enough apart (on the order of light days), then we can roughly approximate an S-type orbit as a single-star orbit, which gives us the Goldilocks zone. If the stars are much closer (on the order of a light hour or less), then it's unlikely that an S-type orbit would be habitable. However, a P-type may be habitable.
A complication here is that we haven't found any non-gas-giants in a P-type orbit. This may simply be a limitation of our current technology. Some astronomers propose that moons of the gas giants could possibly be habitable.