Are continuous densities only good approximations? In a number of situations, continuous densities (e.g. for mass, electric charge, etc.) are used in calculations. However, for example, as far as I know, matter is not continuous. Also, for example, electric charge is discrete. So, are densities only good approximations?
More generally, is there a reason to expect that using the mathematical analysis tools (based on the continuum) like derivatives, integrals, etc. will give theories permitting us to make accurate predictions in the discrete physical world?
 A: Perturbation theory is the thing you're going to be looking for, but because you're on the topic, I want to correct something.  We do not know that matter is not continuous, nor do we know that electric charge is discrete.  Science doesn't actually make that claim.  What it does claim is that the behavior of the universe is well modeled by such discrete models.
Which means what you're really comparing are not a model vs. reality, but rather a model vs. another model.  In practice, we find that the discrete models do a much better job of predicting behaviors at the smallest granularities.  In fact we find this to be true to such a degree that we often just handwave away that little bit of modeling and say "Electric charge is discrete."  However, I draw attention to it because your argument is "because the world is X, model Y must be false."  In reality, your argument is "because we find model X to be effective, model Y must be false because its different from X."  Countless scientists through the ages have been ham-stringed by believing that the prevailing model of the times was "correct" and thus went off in the wrong direction!
Anyways, that aside, we do find that the continuous models we learn about in most of our physics classes are indeed consistent with the fancy discrete models we learn in higher level classes, as the number of entities gets very large.  As an excellent example, consider the blurry line between quantum physics and classical physics.  Take a grain of sand.  In theory, its behavior is very well modeled using quantum mechanics.  For each electron in each atom we can assign quantum states, and show its behavior with a great deal of certainty.  Of course, there's about 50 quintillion atoms in a grain of sand, and we really don't have time to measure all of their states.  Instead, we use the central limit theorem, and assume that the different states are independent.  With CLT in play like this, the quantum effects are reduced to a tiny sliver of effect so tiny that it is completely impossible to detect.
We can show that such systems work this way using perturbation theory.  In perturbation theory we assume some coarse structure and then see how perturbations might affect the outcome.  For example, we may look at the question "what if we were wrong about just one quantum state in the grain of sand... how would it affect the properties of the grain?"  With perturbation analysis we can quickly show that the effects of this discrete model "disappear" into the noise.
In the end, what we find should not be so surprising.  The effect of being wrong about such minor quantum effects is so trivially small that we can't even detect it, except for in specialized experiments whose sole job was to set up the test subject such that it indeed was dependent on those little key details.  Quantum eraser experiments, for example, have detectors which are explicitly constructed to be as sensitive to these non-classical discrete effects as possible.
But they're all models in the end.  Always good to remember that.
A: 
In a number of situations, continuous densities (e.g. for mass, electric charge, etc.) are used in calculations. However, for example, as far as I know, matter is not continuous. Also, for example, electric charge is discrete. So, are densities only good approximations?

Yes.
I'd go further - all our theories are fundamentally just approximations to the real world.  The level of approximation depends on the theory, our ability to compare it to the real world through experiment, and the purpose we intend for the theory.
Continuous density functions are approximations that enable us to make useful models of large systems.  More detailed models would rapidly become unworkable and if our models are reasonable we can get sufficient accuracy.

More generally, is there a reason to expect that using the mathematical analysis tools (based on the continuum) like derivatives, integrals, etc. will give theories permitting us to make accurate predictions in the discrete physical world?

That term "the discrete physical world" is an assumption.  Have you any proof it is discrete ?  I would suggest that assuming it to be discrete is an approximation, just as much as anything else is.  In some applications a theory based on discrete concepts may be either more or less useful than a model using continuous concepts.
We don't "expect" mathematical tools to work out.  We employ them in a way that produces useful theories with sufficient accuracy for our needs from that theory.  We hope to be able to match our theories and experiment - we reject theories that do not match experiments they are intended to predict within a given accuracy.
In practice this has produced many useful theories, but also some which prove to be less useful or even to contradict known facts.  Newtonian mechanics is enormously useful in practice, but it predicts things we know now are false.  It is a useful theory.  We accept the limits of it's usefulness and employ other theories ( if we have them ) when we know we are pushing past the reasonable limits of the theory.
Our theories are models based on model assumptions and their usefulness is limited to where those model assumptions are valid.  In general when we step away from the intended range of assumptions, the theory breaks down.
