Can a present participle follow a subject?

They are examples of absolute constructions:

Absolute construction From Wikipedia, the free encyclopedia:

In linguistics, an absolute construction is a grammatical construction standing apart from a normal or usual syntactical relation with other words or sentence elements. It can be a non-finite clause that is subordinate in form and modifies an entire sentence, an adjective or possessive pronoun standing alone without a modified substantive, or a transitive verb when its object is implied but not stated. The term absolute derives from Latin absolūtum, meaning "loosened from" or "separated" ...

Example Usages:

Being a word, phrase, or construction that is isolated syntactically from the rest of a sentence:

“this being the case, let us go”

"The referee having finally arrived, the game began....

More obviously related is

"The game began,/– the referee having finally arrived."

Usually, the noun group before the -ing form does not inflect for case, but this rare use of inflecting pronouns poses a problem. I'd agree with Leon that it's hard to make a decision between nominative and accusative, with the absolute construction being 'self-contained' and almost certainly elliptical. Informality would argue against using the nominative pronouns here, but then informality would argue against using either alternative:

some days we went on adventures – he would manoeuvre our scooter while I rested my chin ...