If the Arduino is anything like a PIC µC then you have no hope of sampling at 44KHz. Most simple µC have quite a slow sampling rate (like 100's of samples per second).
If you want faster then you'd be looking at using something like a dsPIC which has an actual audio grade ADC in it, or use an audio ADC externally that can send I²S data to a µC that is fast enough to respond to it.
I have done some similar work recently while designing a digitally controlled amp.
I had the output of the first stage of the amp going into an analog input on the controlling PIC to then control a bargraph of LEDs for a simple VU meter.
For an output from a PC soundcard you're probably looking at around 1 to 2 volts voltage swing. For my system I wasn't too fussed about frequency and such - just pure peak amplitude - so I passed the signal through a small shottky diode first to trim off the negative voltages. This simplified my design a whole lot.
I am also designing a small frequency analyzer at the moment, and am looking at having selectable op-amp based band-pass filters based around this design: http://www.wa4dsy.net/robot/bandpass-filter-calc which so far has given quite good results. I am varying some of the resistor values by a combination of digital pots and analog multiplexers.
I would certainly recommend at least protecting your analog input(s) with op-amps to limit the maximum voltage they get - just in case. You don't want a voltage spike blowing up your Arduino now do you? Easier to replace a blown op-amp.
And as for a signal for testing? There are many free signal generators for the PC available for download if you do a little google for them. They will let you select waveform, frequency, amplitude, phase, etc. Even allow summing of waveforms to give new waveforms if you're lucky.
You can even use your PC soundcard as a rudimentary scope as well with the right software and a small home-made probe. There is software and designs around for this too on the net.
Oh, and remember to isolate different stages / voltage levels with capacitors in the audio signal. As a rule of thumb, if I am changing PSU voltage levels, I always introduce a capacitor to isolate the stages. So, I had one on the input signal, one on the stage 1 -> stage 2 (+/-5V to +/-12V power supply), one on the stage 1 -> analog input, and one again on the output. It pays to take no chances with stray DC offsets wandering into the wrong part of the circuit.
If this is indeed an issue of analog/digital separation there are a number of electrical strategies to overcoming it. The easiest of these is to have a separate ground plane for your analog circuitry from your digital cicuitry and join those ground planes at the common return (i.e. the power input to your board). This is called a star-ground topology. I'm not sure what you're going for with the diodes you mention in your question, but you might also consider filtering the VCC to your analog circuit through a ferrite bead and capacitor, and make sure you have adequate decoupling capacitors near all the VCC pins of your amplifiers and other analog circuits.
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Without having ever used any of those parts, let me see if I can take a stab. Do you happen to have access to an oscope? If so you might want to check your signal before you start building anything.
Most likely your microphone/amp is outputting a wave that is centered around 0v, meaning you have + and - voltages. Think of a sin wave that fluctuates between -1v and 1v. In order for your micro to use this, you will need to add a DC offset such that you most negative voltage becomes slightly above 0v and your most positive voltage is slightly below the max that your micro can read (probably around 5v).
With out looking into your components in more, it is hard to tell you specifically what you need to do in order to get your DC offset, but maybe this will put you in the right direction.