Should all this be feasible?
Yes... But as a 1st project, I'd suggest you start by making an LED get brighter/dimmer with the volume (you can use the pin-13 LED built-into the Arduino board). Then, try adding a motor driver so the speed gets faster when the LED gets brighter.
Take it one step at a time... When something is done, then you can try adding more features until it does everything you want.
When you're writing your code, take the same approach adding one or two lines of code at a time, test-compiling and test-running as you proceed. And as jremington recommended, make use of the serial monitor so you can see variables change, and you can send out messages about what the code is doing.
I am looking to have sound input(could be analog or digital)
I don't think you'll be using digital.
Digital requires a specific format/protocol and the associated electronics, etc. USB? S/PDIF? HDMI?
I am looking to have sound input(could be analog or digital), specifically amplitude, control the speed of rotation of a small motor at very low RPM's, like between 4 and 12RPM's with no significant load on the motor....
...to a DC or PWM motor shaft.
You can control the speed of a DC motor with PWM. But to go that slow you'll need a geared-down motor (or a big load on a small motor). And, you'll need a motor-driver circuit because the Arduino cannot directly power a motor.
I would simultaneously also like to have the amplitude of 3 specific frequency ranges(ie. Lows, Mids, and Highs), trigger 3 discrete relays when a predetermined amplitude threshold is reached in a given frequency range.
The easiest way to handle that is with the [u]MSGEQ7 chip[/u]. It gives you 7 frequency bands, so you can either ignore some bands, or combine bands in software.
Or, you can do it entirely in software with FFT/FHT like the spectrum analyzer example.
The project is a laser light show, where 3 different 30mw 3v colored lasers...
...Each of the three lasers will only illuminate when it's predetermined frequency range reaches a certain amplitude threshold.
I'm not a LASER expert, but usually LASERs require a special power supply with additional special circuitry if they are to be "flashed" on & off. But, maybe low-power (30mW) laser diodes can be powered just like an LED... I don't know...
...but nothing that connects sound amplitude detection with variable motor speed, or frequency detection/analysis with the activation of relays.
That's something you'll probably have to figure-out yourself. Work-through and study some of the basic [u]examples[/u] and try to understand what the software is doing. Read-through the [u]language reference[/u]. If you are new to programming you won't understand everything in the language reference, but it will give you an idea of what you can do and it doesn't take that long to read-through.
The two most important concepts in programming are conditional execution ([u]if-statements[/u], etc.), and [u]loops[/u] (doing something over-and-over, usually until some condition is reached).
Once you understand those two concepts, you should have a basic idea of how to design and develop a program that does something useful.
...plus an analog microphone and amplifier that outputs a 2-7V sound signal.
Since the Arduino cannot handle the negative half of the AC audio waveform, you'll need to [u]bias the input[/u] (two resistors and a capacitor).