"No need for A/C, as that compressed air coming out is going to be cold"
Actually compressed air is hot !
How do you think a diesel engine gets fuel ignition inside its cylinders? Super heated air achieved through very high compression.
The act of compressing air will increase its temperature. The air would not stay at elevated temperatures for terribly long, though. Most likely the compressor use some means to cool it before putting it in the tank (since the heat energy the air contains is at that point worse than useless).
The act of uncompressing air will cool it quite significantly. Actually, the one useful application I could see for a compressed air tank would be as an air conditioner for an electric car. I wonder how much air would be required.
It warms up when it is compressed, but cools when it is decompressed, i.e., released to atmospheric pressure.
As the air is compressed it will get hotter. The heat will conduct away through the storage container walls until ambient temperature is reached. When the room temperature compressed air expands (adiabatic expansion) it will get VERY COLD.