Learn Finnish: Mies nukkuu vaunussa kyynärpäät pöydällä. - The man is sleeping in the carriage with his elbows on the table.

This is an absolute phrase of state (irtonainen tilanilmaus). In Finnish, you can describe a physical posture by simply pairing a body part or clothing item in the nominative case (kyynärpäät) directly with a location word or local case (pöydällä). You don't need a word for "with", and trying to use kanssa here would sound unnatural.

Why doesn't kyynärpäät have a possessive suffix to show they are his elbows?

When a body part clearly belongs to the subject of the sentence (mies), Finnish often drops the possessive suffix. In these absolute phrases, the possession is naturally implied by the context, so adding a suffix like in kyynärpäänsä is unnecessary and would disrupt the phrasing.

Why is kyynärpäät in the nominative plural case rather than the partitive?

In this construction, the noun indicating the body part acts almost like a mini-subject for its own little descriptive clause (as if saying "his elbows are on the table"). Because of this subject-like role, it takes the nominative case (kyynärpäät) instead of the partitive, even though the main action of the sentence is ongoing.

Could I use a postposition and say kyynärpäät pöydän päällä instead?

Yes, grammatically you could use the postposition päällä (on top of) instead of the adessive case suffix -llä. However, absolute phrases of state are meant to be snappy and concise. Using the local case (pöydällä) is much more common and sounds more natural for describing everyday postures.