Should I replace both my outdoor condensing unit (which includes the compressor) and the indoor coil on my central air conditioning system at the same time?
In most instances, yes. Matching a new condensing unit with a new coil is the only reliable way to be certain you are going to get the rated efficiency of the new equipment. Matching a new, high SEER (seasonal energy efficiency ratio) condensing unit with an old indoor coil probably would not result in optimum efficiency.
Another thing to be aware of is the government phaseout of R-22 freon. If you’re air conditioner uses R-22 you may want to consider replacing the unit or it may become more expensive to fix within the next few years.