How to set PS for the most reliable image previews and how to make sure the color spaces are the same across plateforms

Question

I've been having an ongoing problem with some of my images changing color/luminance values after merging or flattening an image. This seems to most often happen if I use one of the filters (like crosshatch or notepaper, not sharpen or blur). I can make a copy using command/option/shift + E which most often will look exactly like what's on screen.
Some people have suggested that it's to do with resolution, PS showing me a low resolution image but I don't think so because the histogram changes -- sometimes in a big way, but most often those spikes in the luminance levels are shaved off. Also, when I have the image open at 100% and effect a merge/flatten, the color changes.
I'd like to know if there's a way to have PS show me one for one what I'm working on while I'm working on it, or if there is a way to control the color profiles when I'm saving. My best guess at this point is that there is a color profile mismatch going on somewhere. I just don't know where to look for it.
Typically I work in 16 bit, so my color space is set to ProPhoto RGB; my camera is set to the Adobe color space;
color management policies are set to preserve color profiles;
conversion options --> intent is set to relative colorimetric.

Answer 1

But it is calibrated. I use Xrite 1 and update it every month (on the first).
The colors can be really off. Here's an example.
https://flic.kr/p/r6A1Zd (what I want) https://flic.kr/p/qc5QTD (not what I want)
Also, I can't see how the bit depth will have any effect since the change in color happens in PS when I merge or flatten. It's 8 bit through the editing and 8 bit when I flatten it.
Have you any suggestions on what I can do on the computer side?