In my opinion, we could say "..sleep with your closed/open mouth."
Please explain to me why we use adjective closed/open after the noun mouth? Please tell me the difference between the two usages.
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Saying "You should sleep with your mouth closed" means that when you sleep your mouth should be closed. Saying "You should sleep with your closed mouth" means that sleep is somehow called for due to the fact that your mouth is closed.
With modifiers(**) like open/closed the presence of the modifier before the noun implies that the modifier is describing the current state of the noun -- in effect selecting which possible version of the noun is being referred to. With the modifier after the noun, on the other hand, the implication tends to be that the object identified by the noun should have it's configuration changed to what the modifier describes.
** I use "modifier" because I don't want to get into a discussion as to whether it's an adjective or some form of verb.