32-bit (linear) .HDR in in PS and save, the save operation will basically ruin the .HDR because PS clamps pixel values at some threshold.
I have tested this and never found it to be the case in normal usage. I understand where this sentiment comes from, for example PS's 16bit mode is actually 15bit (and shifted?) because they wanted the operations to be faster... 20 years ago and the legacy is still there.
While it might cut some values off, I believe those values are far out of any usable range for Sky domes or Displacement values. It definitely doesn't affect any Sky Domes I've tested.
But it's better to use .HDR than .exr if you have truly large dynamic range (HDR supports up to 256 stops).
But that is talking about dynamic range of floating point linear files. 8 & 16bit Integer files have zero dynamic range. The unsigned values are simply 0 to 255 and 0 to 65635.
So if Photoshop cuts off some values at 16bit range, we're talking unperceptible difference in precision.
Everything Pokoy wrote is correct, but default dot gain is indeed 20 perc. you can reinstall Photoshop or simply choose North American General Environment to check :- ). Not that it affects anything you ask about.
And yup, the larger issue is baked in gamma correction. But that has nothing to do with Photoshop. Using integer files with baked gamma was simply never good choice for anything outside of pure B&W masks.
If you need to modify these types of files, convert them to 32bit environment in Photoshop first (even if you plan to save to 16bit .exr, since that is associated with 32bit linear environment only with half the precision) and then make any changes.