The heightmap looks like that because painter uses signed values for height. What would be 0.5 in designer is 0 in Painter. It's just a visual thing, your height should be fine.
I've no idea about the displacement as I've never tried it
Painter could show the displacement too . it's in shader settings.
By default any height texture is rendered with kind of mild influence to pixel normals . Sometimes hardly visible. No connection to real height amplitude it supposed to be.
I use" height to normal" filter put into normal slot of a layer with same input height texture. Did it from SD normal node. Have no idea what default one, more advanced is doing. As usual in substances you see lots of sliders and no idea what they are all about, no hints, examples or explanation in help. So it's really simpler to just make your own.
Same with AO filter .
Would love to see some checkbox in SP that would nomalize height channel preview to visible 0-1 in view-port only.
Also it seems Painter does all the pixel calculations in output resolution while SD could do it the way whatever you want.
I usually work in twice bigger resolution and scale down at export.
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I've no idea about the displacement as I've never tried it