Hi Neil,
Welcome to PixInsight Forum.
1) For the Gaussian PSF, where are the settings determined? Is it just guessing, or where/how is the PSF created?
We still don't have an officially released PSF measurement tool (although this will change soon). As has been noted in other posts though, guessing a very good PSF is usually quite easy by some trial-error work. With a little practice it's a matter of a few minutes.
Always start by looking at some of the smallest stars in your linear image (you need the STF tool to do this). As a star is a point source, it provides a good sample of the PSF. For example, for your image, a standard deviation of 1.2 pixels seems quite appropriate (more on how I know this later).
2) Ditto for shape and aspect ratio?
For deep-sky images, always use pure Gaussian PSFs (shape=2). Other shapes are useful to simulate close-to-diffraction-pattern PSFs, which can be used with high-resolution lunar and planetary images. Unless you have elongated stars that you want to fix with deconvolution, leave aspect ratio with its default value of one (perfectly round stars).
3) After doing the decon on the linear data, I did a histogram stretch and saved the before and after as a 16-bit TIF so I could take into ImageReady to do the animated GIF. The images looked posterized, as though they were 8-bit images. I checked in Photoshop, and they had the same appearance. What did I do wrong?
Nothing wrong from what you've described. A 16-bit TIFF image is pretty standard and should be readable without problems in other applications. As I don't have the applications you have mentioned I can't say for sure, but it seems a problem related to poor screen rendering. I have downloaded the 16-bit images from the links you've posted and they show perfectly in free software applications such as Gwenview and Gimp.
I think you can deconvolve your image further. I have made a small test that you can see in the attached screenshots.
Since you have posted a nonlinear (stretched) image, I can't test deconvolution directly on it. However, I have applied a histogram transformation to linearize the image. The HT has a midtones balance value of 0.98. Of course, this is just an approximate guess but it allows me to show you how to deconvolve images like this one.
The other screenshot shows the Deconvolution instance that I have applied. In general, always use the regularized Richardson-Lucy algorithm with deep-sky images. Regularized Van Cittert works better for lunar and planetary images, but RRL is more controllable for deep-sky.
Note that I haven't used a deringing support image (no local deringing) since this is a quick try. In the real world local deringing is necessary to prevent rings around the brightest stars --
only for these; the rest of ringing problems are fixed by the global deringing algorithm.
When the PSF is very small (as in this case where sigma=1.15px), better results can usually be achieved by upsampling the image prior to deconvolution. For example, you can upsample x2 with the Resample tool. With an upsampled image the PSF is also proportionally larger, and this allows us to sample it much more accurately.
Hope this helps.