Well to get good data, make sure you use good anti aliasing filters.
Also, learn which kind of filter you want to use, if you want accurate impulse response you will want a different type of AA filter then butterworth topology.
A sound level meter will probobly give you a RMS output. You can calculate RMS values from FFT data, or you can measure them with another instrument, theoretically RMS values might be a bit more accurate and the electronics are simpler/cheaper/possibly more stable, but for something like dampening a room to human hearing levels I would not worry.. the accuracy is low. If you were interested in doing something like dampening a precision mechanical/laser laboratory from noises and vibration obviously the answer would be different, but if its a human dwelling you don't need to worry about it too much.
Also, RMS is a bit easier to calibrate, but I think relative levels are adequate for your task.
The data collected using non-microphone sensors may be interesting to investigate things or form useful models, but ultimately you are interested in what you persieve with your ears, I think the microphone is the sensor most similar to your ear drum. Those sensors are more for making sure machinery is not destroying itself etc, since it is actually telling you what is happening to the metal (i.e. how much energy is in the vibrations of the structure of the machine) rather then how much energy is being coupled into the air (which is what you end up hearing).
They might however be interesting to use for lower frequencies (i.e. like if your bed rumbles when a truck drives by, using a accelerometer connected to the leg of the bed might be more useful then trying to use a microphone).
Look up how an oscilloscope measures RMS digitally to understand how the various values are calculated.
Doing your own FFT works well, but the main problem is input amplifier noise and dynamic range. The ideal instrument for this situation is known as a dynamic signal analyzer which is a special instrument that works like a FFT spectrum analyzer (and often has a Voltage/Time interface like an oscilloscope), but it is designed with good anti aliasing filters, only goes up to some KHz (maybe 100KHz) and has an excellent noise floor, dynamic range and various useful measurement functions built in.
I think you will be interested in the 'waterfall' function for doing this kind of testing.
But again, you will want to research good anti aliasing filters for connecting your own sensors to ADCs, they are important, especially for FFT.