Question

Not having studied General Relativity, I have sometimes been puzzled by references to the behaviour for "classic" black holes

Answer 1

The ability to "avoid the singularity" is generally regarded as a special property of the very special, stationary, exact solutions we know for black hole space-times. It has to do with the analytic continuation of the solution of Einstein's equations beyond the region that one can predict based on causal principles. (Basically, if you only know that there is a black hole, and you only know what goes on outside of the black hole, you cannot predict what happens inside the black hole; if you actually see the black hole form, on the other hand, you may have a chance at making this prediction.) So one shouldn't take that possibility too seriously. (I for one won't bet my life on it and jump into a charged black hole.) In fact, one interpretation of the Strong Cosmic Censorship conjecture is precisely that for generic black holes, the singularity is unavoidable once you entered the event horizon.

The bound on the charge-to-mass ratio is, at the present day, more of an imprecise conjecture than a stated fact. There are several problems with that statement: in a dynamical space-time, mass-energy can radiate. So the definition of the "mass" of a black hole is already problematic. (This is also related to the fact that mass in general relativity cannot be defined locally; though there are a lot of work put into definitions of quasilocal mass.) Similarly, the charge of a black hole, in a general dynamical space-time, is not well-defined. What we do know is that we have a three parameter family of exact, stationary solutions to Einstein's equation depending on M,A,Q. Because these solutions are all stationary, the mass and charge are well-defined. Because these solutions are all axisymmetric, angular momentum is well-defined. And within this family we know that were the charge Q to exceed the mass M, the formula that gives the expression of the metric tensor still makes sense as a solution to Einstein's equations, but the formula will lead to solutions with no event horizons. So we conjecture that this is a general fact, despite not knowing how to define the mass and the charge of a generic black hole. Now, there are some cases where this conjecture is known to be true for dynamical solutions. For example, in spherical symmetry, if we also assume that we have, in addition to the electromagnetic field, some other "good" uncharged matter fields (this makes the electromagnetic field non-dynamical, but the gravitational field is still dynamical), then we can prove such a statement using the Hawking mass of the black hole. These types of statements are related to Penrose-type inequalities, most of which are conjectural and only a few have been proven to hold generally. (Remark: there's however evidence that one cannot start with a subextremal black hole and then ``supercharge'' it.)
The horizon area of charged black-holes are smaller than that of uncharged ones. (Again, because of the difficulty in definition, interpret the above in terms of the known stationary black holes.) Your last statement in the third paragraph is incorrect.
The answer to your general question about the interplay and the mechanism is: "no one really knows". The problem is that general relativity, unlike classical Newtonian mechanics coupled to electro-magnetism, or even special relativistic mechanics coupled to electro-magnetism, is a highly nonlinear theory. In classical electrodynamics, the linearity of the system allows you to pin-point the contributors to the dynamics: you can say that the total force acting on this particle is a sum of the gravitational force plus the Lorentz force plus this-and-that. In GR, because of the non-linear feedback, it is in general impossible to disentangle the sum of the parts from the whole. (While the equivalence principle tells you that locally in inertial frames stuff behave as it were linear, the sort of questions you were asking necessarily involve long range effects of gravity and electromagnetism.)

To summarise: there's still too much we don't know, even with regards to the basic definitions, in general relativity to be able to answer your questions. We don't have a completely satisfactory definition of black holes that can be used locally (as oppose to teleologically), and we don't know what the local definitions of mass or charge should be. We certainly don't have a general description of how black holes should behave under the influence of electric charge. What we do have is a rather limited zoo of examples. This dearth of data points means that a lot of different conjectures can be made to fit those data points It is hard enough to know even which of those conjectures are right, nevermind to try and understand the principles behind such behaviour.