The life of an activated carbon filter is dependent upon several factors: what contaminants you are trying the remove, their concentrations, and the flow rate through the filter. If you are just aiming to remove chlorine and/or chloramine, those compounds are destroyed through a reaction with the carbon. In that case, the carbon is actually consumed. If you are trying to remove volatile organic compounds or those 'pond scum' tasting compounds, those contaminants are adsorbed onto the carbon matrix. In either case, flow rate has a strong bearing on how quickly a carbon cartridge no longer does its job and the contaminant makes it through the cartridge into your supposedly clean brewing water. Slower flow is always better at removing any contaminants. In the case of the typical 10-inch cartridge, chlorine removal is maximized when the flow rate is less than 1 gallon/minute. For chloramine and pond scum contaminants (MIB or geosmin), the flow rate needs to be less than 0.1 gallon/minute for good removal. The feed rate to the typical home RO system is at that very low flow rate. That is why activated carbon works as a pretreatment for RO feed water, the very low flow rate.
You should periodically check the contaminant removal of your carbon filter. If you are aiming for chlorine or chloramine removal, then a simple Total Chlorine test kit for swimming pool use can be adequate for checking if those contaminants are breaking through the filter. For the other contaminants, you may be forced to give the water a taste test or you can send it in for a costly lab test. Tough decision.
With regard to storage, either wet or dry is probably OK. However, since microbes like wet environments, storing the filter wet MAY invite more microbes to grow in your cartridge than dry (I guess it would actually be moist) storage may produce. In any case, you should consider the output of any carbon system that is periodically disconnected from the water supply to be infested with microbes and you probably shouldn't drink that water. The good thing is that the brewing process sanitizes all that potentially-infected water and your beer will be safe.
