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First off, if you are buying this iron to do electronics work, do not buy a gun-type iron.  They are primarily used for larger, heavier connections that soak up a lot of heat.  One of those bad boys would peel the traces off a PCB before you could say "what's that smell".

Your best bet is a soldering pencil in the 25 to 40 Watt range.  If at all possible, avoid off-brand or Radio-Shack brand irons.  They are typically referred to as "wood burners" by serious electronics guys.  Their temperatures fluctuate wildly during use and they are cheaply made and thus will not last long.  Temperature control is almost a necessity, but BitPop's temperature-fixed iron will definitely be better than no temperature control at all.

Personally, my favorite iron is the Weller WESD51 Soldering Station.  Definitely the most comfortable iron I have ever used.  Of course, the only irons I've used are it and a 35W Radio-Shack wood-burner.  It's $130 at Amazon.com, and unlike the wood-burners, it should last you for years if you take care of it.

I'm not quite sure yet, really.  I want to make something myself, but not being a chip musician, I don't really know what you guys really want or need.  I did find this open-source MIDI sequencer that may be useful.

I love this question!  I've been learning some of this in my Digital and Analog Communications class.

This is a perfect example of what is called the Gibbs phenomenon.  It's what happens when a square wave is passed through a low-pass filter.

In the above picture, we see the waveforms of two low-pass filters.  Any signal that falls underneath the large peak at the center of the graph is allowed through.  Ideally, no other wave component is allowed to pass, that is all waves that fall outside the central peak are filtered out.  However, you can see where the low-pass filter still exists outside the main filtering frequency.  Higher frequencies that we want filtered out are still slipping past, generating the diminishing sine wave you see at the top and bottom of the square wave.  As we make the low-pass filter narrower, the Gibbs phenomenon also gets smaller, but can never be made to disappear.

Basically, it's a technical limitation and a fact of life that really has no effect on you unless for some bizarre reason you need a completely clean square wave.

You can also see the Gibbs phenomenon in your DMG square wave there.  Notice how the wave seems to extend beyond it's logical point and ripples ever so slightly before settling back down into the primary wave?  The DMG square wave appears to be on a very narrow low-pass filter, resulting in a very small Gibbs phenomenon.

EDIT: additionally, the Gibbs phenomenon can be seen in the generation of square waves themselves.  You can generate a square wave by combining multiple sine waves together, with the result being a square wave exhibiting the Gibbs phenomenon.  The greater the number of harmonics (the number of sine waves you are combining), the smaller the Gibbs phenomenon.  If you want, I can try to teach you the math, but I'm afraid I barely understand it myself.

Actually, this batch is not yet done.  I still have 8 more here on my workbench.  Problem is I ran out of mounting screws to fit the PCB inside the case.  Hopefully I will have the parts to finish them in a week or two.  Meanwhile, all 17 completed ArduinoBoys have been sold and will be shipped to their new owners in a couple days.

Only three left.  Don't worry, though.  I will be making more.

Also, those photos you guys wanted to see.  Sorry it took so long, but my camera takes horrible close ups.  Be glad that my roommate's camera phone doesn't have the same problem.

That has got to be the largest free-form circuit I have ever seen.