(something like this), using a microphone...
...Would I need some extra controllers for this project?
That particular LED strip has built-in controller & driver chips, so you only have to supply power and you can control it directly with the Arduino. But, the specs are incomplete. I suggest you buy from [u]Adafruit[/u]. They specialize in this kind of thing, and they have complete specs and hardware/software examples.
Do you want individually addressable LEDs? ...Of course its up to you, depending on what kind of effect you want and individually addressable LEDs can be more "spectacular", but with amplitude and frequency you may only need intensity & color.
I have found several microphones such as this ones.
Do you really want to use a microphone that will pick-up all of the sound in the room, or would you rather tap-into a line-level or headphone-level signal?
Can I get both frequency and dB input from this microphones?
Of course a "plain" microphone will convert acoustic level & frequency into electrical level & frequency. But, the signal from a microphone is only a few millivolts and it needs to be amplified. Also, electret microphones have to be powered, and the Arduino can't read the negative half of an AC audio signal, so it has to be biased. The microphone board should take care of all that.
The specs on the microphone board you linked to are also incomplete. Someone answered a question saying it has an "analog" output, but it's not clear if that's an audio signal or a proportional varying-DC signal, so that board might loose the frequency information.
Again, I recommend buying from an electronics supplier that provides complete specs such as Adafruit or [u]this one[/u] from SparkFun. The one thing I don't like about the Sparkfun board is that it doesn't have adjustable gain/sensitivity. (You can adjust the sensitivity of the software to some extent.)
Once you have a signal (from a microphone board or from a line/headphone output) there is a nifty chip called the [u]MSGEQ7[/u] that gives you 7 time-multiplexed DC signals proportional to the amplitude of 7 frequency bands. If you don't need more than 7 frequency bands that takes a big load off of your software. Otherwise there is an FFT software library for finding the frequency information. But, I have a feeling that with neopixels (individually addressable RGB LEDs) your software is going to be "very busy" sending-out the LED information.