4. A digital to digital signal chain does degrade
"When I use a digital source, I get a pure digital-to-digital signal chain using HDMI."
This is true in an essentially meaningless sense, and untrue in the sense in which most people actually understand it.
The assumption behind that statement is that the signal flows, unaltered and without degradation, from a digital source to a digital display without ever being converted, and that by eliminating these conversions – specifically, digital-to-analogue conversions – one gets a better picture. But the HDMI signal is not the same as the signal recorded on a DVD, or sent in an ATSC or QAM transmission; all of those are compressed formats which encode video in an entirely different way from HDMI. Accordingly, to get from the one to the other requires decoding and conversion. In every case, the signal is decoded and rendered as a video stream. If the original signal is in one resolution, and the output format is in another, the image will be rescaled; if the original signal is recorded in one colourspace and the output format is in another, it will be converted. There is nothing inherently perfect or error-free about digital-to-digital, as opposed to digital-to-analogue, scaling and conversions, and some things (scaling, in particular) are often more easily done well in the analogue domain than in the digital domain.
So, yes, a DVD player putting out an upscaled HD resolution through an HDMI cable into a plasma display is an "all-digital" signal chain – but it's an all-digital chain in which the colourspace is being converted, the original signal is being decoded and converted to another format, and the image is being rescaled not once, but twice along the way. Is doing this digitally superior to doing it in a chain that involves analogue conversions? It may be, or it may not be. But there's no reason to think that it necessarily will.
"Because HDMI is a digital signal, it doesn't degrade when run over a long distance like an analogue signal does, because it's just ones and zeros."
Not true at all.
Firstly, it is true that if a digital video signal stays intact from one point to another, there's no degradation of the image.
The digital signal itself can degrade, in **electrical** terms, quite a bit over a distance run. But, if at the end of that run, the bitstream can be fully and correctly reconstituted, it doesn't matter what degradation the signal suffered – once that information is reconstituted at the receiving end, it's as good as new.
That's a big "if," however. Ideally speaking, digital signals start out as something close to a "square wave," which is an instantaneous transition from one voltage to another; these transitions signal the beginnings and ends of bits. (In practice, such transitions aren't strictly possible, and trying to achieve them can generate harmful noise; consequently, high-order harmonics are usually filtered out which results in the wave starting out squarish but not-quite-square.) A square wave, unfortunately, is impossible to convey down any transmission line because it has infinite bandwidth; to convey it accurately, a cable would have to convey all frequencies, out to infinity, all at the same level of loss ("attenuation"). What happens, therefore, in any run of cable is that a digital signal starts out looking relatively nice and somewhat square, and comes out the other end both weaker and rounded-off. The transitions that mark the edges of bits get smoothed and leveled to the point that, far from that ideal square wave, they look like relatively gentle slopes. Portions of the signal lost to impedance mismatch bounce around in the cable and mix with these rounded-off slopes, introducing an unpredictable and irregular component to the signal; crosstalk from the other pairs in the HDMI bundle also contribute uneven and essentially random noise. As a result, what arrives at your display doesn't look very much like what was sent.
Up to a point this won't matter; the bitstream gets accurately reconstituted, and the picture on your display is as good as the HDMI signal can make it. But when it starts to fail, it starts to fail conspicuously and dramatically. The first sign of an HDMI signal failure is digital dropouts – these are colloquially referred to as "sparklies" – where a pixel or two can't be read. When these "sparklies" are seen, total failure is not far away; if the cable were made ten feet longer, there's a chance that so little information would get through that there would be no picture on the display at all.
The shame is that, with HDMI, this is prone to happen at rather short lengths. When DVI was first introduced (same encoding scheme, same cable structure, but a different connector from HDMI), it was hard to find cables that were reliable in lengths over 5 metres. Today, a good HDMI cable can be relatively reliable up to about 15–20 metres, but because different devices tolerate signal degradation differently, it's impossible to say categorically that a 20m cable will work; it's only possible to say that it will work with most devices.
Why is that? Well, it all has to do with design.