We investigate cold rubidium plasmas in a particle trap that has the unique capability to simultaneously laser-cool and trap neutral atoms as well as to confine plasmas in magnetic fields of about three Tesla. The atom trap is a high-field Ioffe-Pritchard laser trap, while the plasma trap is a Ioffe-Penning trap that traps electrons and ions in separate wells. The observed plasma dynamics is characterized by a breathing-mode oscillation of the positive (ionic) plasma component, which feeds back on the behavior of the negative (electron) component of the plasma. At higher densities, the observed oscillations become nonlinear. The electron component has been found to undergo rapid cooling. We further report on the recombination of magnetized plasmas into Rydberg atoms in transient traps and quasi-steady-state traps. In transient traps, large numbers of recombined Rydberg atoms in high-lying states are observed. In quasi-steady-state traps, the measured numbers of recombined atoms are lower and the binding energies higher.