Given what is known about what Jake believed and espoused, the problem might more aptly be described as him representing his movement too obviously.
Shouldn't it be "his representing his movement too obviously"?
Honestly I'm pretty confused about this stuff
I would have written "his". "Him" is a clinker to my ear.
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Yes, you are right.
The +ing form of the verb is the present participle. It is used to make the progressives like "You are running." Additionally, the present participle can be used as an adjective as in "Running down the street, he fell." Another use of the form is called a gerund, and that is the one you have here. A gerund is the +ing form used as a noun. In your sentence, "representi