converting/exporting to jpeg will always slightly change the colours
the 'Gamut' alert is to warn you that a color selected cannot be displayed or printed exactly using the values you selected. see https://helpx.adobe.com/illustrator/using/color.html
2 using Bridge, can you use Edit > Color Settings to synchronise your settings between Adobe apps,
Have a look at a similar post (I know it is illustrtor question, but it is similar) that could help: Solved: Hex Codes Changing in Adobe Illustrator v27.0 - Adobe Community - 13302801
Can you explain the goal you are trying to achieve? Why you are exporting to jpg?
Also, make sure the exporting process does not use print simulation, that could try to 'mix the inks'
There is a good reference on color profiles here : Working with color profiles, Adobe Acrobat
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