How actively does your brain physically prune connections? Is pruning defined as neural apoptosis or are connections (neural synapses) simply separated? 

At what rate, if known, does this occur in individuals (ages 18+)?
Apoptosis is the programmed death of a cell. This is not required for pruning of connections.

During development, axons and dendrites of neurons undergo extensive lengthening, branching and also retraction. Simultaneous with this process, many many synapses are being constructed and deconstructed between axons and nearby dendrites. Also during adulthood, synapses are constructed, strengthened, weakened and deconstructed.

Hard numbers:
Between the ages of 20-90, a 10% reduction in number of neurons occurred. On average, neurons make around 7000 synapses per neuron. Under the assumption that the number of synapses per neuron doesn't change, there will be around a 10% reduction of synapses between ages 20-90. The paper referenced below states that estimates of numbers of synapses over different ages is in progress, but I couldn't find a follow-up publication.

Source: Pakkenberg, B., Pelvig, D., Marner, L., Bundgaard, M. J., Gundersen, H. J. G., Nyengaard, J. R., &amp; Regeur, L. (2003). Aging and the human neocortex. Experimental Gerontology, 38(1-2), 95-99. doi:10.1016/S0531-5565(02)00151-1