Why do Dyadic filterbanks downsample the high pass signal portions?
[Good question, that made me rethink of stuff I believed natural. I shall incorporate them in future lectures]

Downsampling the highpass (and the lowpass) provide you with a critically sampled filterbank, in other words, not redundant. Critical sampling is a price to pay in some applications like compression, and in the context of finding an orthogonal transform. So yes, this is a limitation: one should take care of aliasing, and of the imperfect character of the designed filters. And yes too, filterbank coefficients are subject to folding/mirroring. If one were to analyse these coefficients as if they were a genuine signal, for instance with linear filters or Fourier transforms, that could raise some concerns. 

But remember that the DFT can be interpreted as a filter bank as well, downsampled by the length of the signal (plus zero-padding). Each DFT coefficient becomes, in this interpretation (correct me if wrong), a single "sample", rejected to a baseband. 
I understand your concern, but it is very rare to apply a linear filter to highpass subbands. On the contrary, one usually employs nonlinear filters and selection tools. 
Would you perform linear filtering or Fourier analysis on it without caution? Of course, if you keep this coefficient, set the others to zero, and inverse the DFT, you get a signal back, a sine, and then you could perform traditional processing on it.

The same applies to FIR (even IIR ones) filter banks. Here, the downsampling is much smaller. And the series of coefficients longer. But in many cases, these coefficients are not "signals" per se. Very often, they are not processed "linearly": they are cancelled, shrunk, thresholded. Most of the work focuses on both amplitudes AND  location, due to the relative locally of the filters (contrarily to DFT).

If you want to keep with the linear interpretation, the best is probably to set the other coefficients to $0$ and reconstruct with the inverse FB. Then, you have a real signal back again, and you can safely Fourier, filter, etc.

But not all wavelet representations do that: "I notice in all of the visual representations" and "why does every representation insist on downsampling the high pass subband?". This is not always the case, and you have a wealth of oversampled, shift-invariant, etc. filter banks, not only dyadic, that do a much better job. 

For instance, you can avoid the downsampling on the highpass (and also on the lowpass), and you end up with a more redundant representation, that can be useful for some processing cases. Then, you can process the subbands more normally. 

For some (personal) background on (oversampled or not) filter banks, which content I can speak of: in 1D Optimization of Synthesis Oversampled Complex Filter Banks, in 2D A Panorama on Multiscale Geometric Representations.