You briefly accelerate a charge for some time, and it will both radiate electromagnetic energy and gain kinetic energy.
The kinetic energy it gains seems to be governed by Newton's Second Law, so by putting some amount of work into the system, somehow we get that amount, plus a little bit more, of energy out?
No, it doesn't violate the conservation of energy.
The energy emitted as electromagnetic energy by the charged particle is lost from the kinetic energy of the charged particle. Thus, the total energy input (via the work by the force accelerating the charge) to the system of the charge and the field is split between kinetic energy of the charged particle and the field energy.
See, for example, bremsstrahlung.
Nope!
The kinetic energy it gains is the amount of work you apply to it minus the energy it loses to radiation. So it gains less kinetic energy than you would expect just from Newton's Second Law applied directly to the force you apply to it.
