The answer to this question is not simple. A charge is surrounded by an electromagnetic field. Whether or not that field is "radiating" depends on the behaviour of that field as one gets very very far away from that charge. The radiation part of the field falls off as one over the distance from the charge, while the standard Coloumb field falls off as one over the distance squared.
If the charge is constantly accelerating everything becomes ambiguous since the charge itself eventually attains the speed of light and races off to infinity. The standard statement is that if the charge is in constant acceleration then there is no radiation. This is supported by the fact that the self force on the electron ( the force due to its own electromagnetic field) goes as the time derivative of the acceleration, not as the acceleration. Ie, the radiation depends on the change in acceleration.
The question becomes more complex in a gravitational field. A gravitational field is a change in the concept of distance in space-time, and that change in the metric (distances) of spacetime itself affects the elecromagnetic field surrounding the particle. In that case of a static electron in a static metric ( one that does not change in time) there is no radiation at infinity even if that electron is "accelerated" ( i.e. does not follow a straight line in that spacetime). For example, an electron held still on the surface of the earth is accelerated upward-- since the natural straight line is a path toward the center of the earth. An electron following a straight line ( that is: dropped near the earth) is not locally accelerated but is radiating. Note that the wavelength of that radiation is very very long. For example, for an electron dropped on the surface of the earth, the radiation has a wavelength that is of the order of the distance of the earth to Mars.
The subject becomes even more complex since changes in the metric far from the charge can interact with the field of the charge so that radiation is produced. Also a charge that does not radiate ( e.g. is stationary on the surface of the earth) will feel a self force, a radiation reaction force, due to the changes that the gravitational field produces on the electromagnetic field.
The first part can be answered fairly simply, if not very rigourously. For that you'll need to consult an Electrodynamics textbook or take a 4th year university Electronics and Magnetism course. An electric charge can be thought of as "emitting 'lines' of electric field" that spread out radially in all directions. Now, if you are another charge sitting some distance away, the field 'line' from the original charge goes by you in a direction given by the straight line from the first charge to your position. So far, easy. But now suppose the "emitting" charge MOVES. Then the direction may change. Also the strength of its electric field at your poisition, if it gets closer. There are two possibilities: either you know INSTANTLY that the other charge has moved, or it takes a while for that information to get to you. The latter is in fact the case. This immediately has consequences: if the other charge is moving uniformly, all that happens is that its field at your position is either tilted a little or getting stronger or weaker, or both, because what you see is actually the field from where it was a little earlier; this is difficult to calculate, but you can easily get the general idea. So no big deal. But if the emitting charge CHANGES its motion (e.g. accelerates or oscillates up and down) then the electric field 'lines' it emits get "kinked" as a result. And then another principle kicks in: a changing electric field generates a magnetic field (Maxwell's correction to Ampere's law) perpendicular to the electric field; and if the charge oscillates then that magnetic field also oscillates, generating (according to Faraday's law) a perpendicular electric field, which in turn.... These mutually co-generating fields are what makes an electromagnetic wave, hence an accelerated charge RADIATES.
For the second part of your question, I must confess, I don't know the easy answer. A quick google search yielded this discussion at Physics Forums but I don't get the impression that those guys have come to any clear conclusions. That doesn't mean no one knows the answer.
Thanks for not asking why the electrons in an atom (which, in a simple Bohr model, are certainly accelerated!) don't radiate away all their kinetic energy and spiral into the nucleus in about a billionth of a second. That's how Quantum Mechanics got started, and I don't think you'd be satisfied with a handwaving explanation for that!