What is the grammar of "CJ Dennis had 2 edit suggestions approved"?

This construction with "have" in American English is most curious. It has the form of a causative construction with main verb "have" meaning "cause" followed by a clause describing the event or condition that is caused. The interpretation is quite. different (though there is often ambiguity with a causative interpretation). The construction is used to bring a person affected by an event or condition into subject position. Your example

CJ Dennis had 2 edit suggestions approved.

is actually ambiguous between a causative interpretation that Dennis caused the approval of the suggestions and another interpretation that Dennis was merely affected somehow by the approval.

The English construction is not totally unlike the Japanese adversative passive.