What are the structures appended to the end of the main clause below?  They
feel like participles, but they have pseudo-subjects.  Is it correct to use
the objective pronouns in these structures?

*I slipped my backpack off, and we sat down at one of the metal tables
overlooking the pool, him sipping his Mountain Dew and me groping for a pen
and paper.

*I hope everyone had a great Thanksgiving!

Thanks,
John

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