Bottom line: First reset all color settings to defaults by picking the "North America General Purpose" preset. These are safe settings that can't go wrong. Never change anything in Color Settings without very good reason.
Next, for PNG always use Save For Web or Export. Check "convert to sRGB" and "embed color profile". Also set preview to "use document profile", not the default "monitor color".
Don't use Save As - it's buggy. If you do, make sure to assign the sRGB profile to the saved file and resave.
Generally - and this is something I'm beginning to see as a serious problem - the Export modules have color management disabled by default. Contrary to "main" Photoshop, these modules do not have safe settings by default. This will cause inconsistencies and cumulative problems if you're not careful.
Never, ever work with untagged images. Anything can happen. Which is why it's also a good idea to keep the notification area (that Gene pointed to) set to document profile. This way you keep track of profiles with a single glance, and can spot untagged files immediately.
I'm pretty sure what happens here is that an sRGB or Adobe RGB file gets the ProPhoto profile assigned. Every time that happens, saturation increases massively (and channels get clipped).