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.

Congratulations on doing a lot of research before posing your question. One further thing that would help even more would be the passage that precedes the quotation, to give a context.

The quoted sentence and the verb had itself are, in fact, ambiguous without such a context.

To have plus the past of a verb (call it a passive past participle) can be used in the sense of getting someone to do something. So we can say:

Queen Elizabeth I had Mary Queen of Scots beheaded.

This means she signed an order for the execution to be carried out. This usage is a possible meaning of the O.P.'s sentence. So on this interpretation,

1. C.J. Dennis had <i.e. brought it about that> 2 edit suggestions <were> approved.

C.J. Dennis is in charge of editing and gets someone to approve two suggestions that some online punter has offered.

Then there is the more probable interpretation. The approval of suggestions (which C.J. Dennis had put forward) were caused/made by someone else. Had means here something like acquired, got, came to have. For example,

C.J. Dennis has just had his PhD thesis on recycling by micro-organisms accepted by the Oxford department of biochemistry.

This sentence is made unambiguous by the addition of the actual agent in the award. C.J. Dennis is the happy beneficiary. So the meaning of the O.P sentence (with my interpretative supplements in angled brackets) is:-

2. C.J. Dennis <suggested some edits, of which he> had 2 <of his> edit suggestions approved.

The wider context, for future reference, would have shown for certain.

So what do online dictionaries say about the meaning of ‘have’? My Merriam Webster reveals an astonishing range of synonyms, depending on context. The verb seems to be able to mean almost anything, depending on context. The best word for this is to call it a (pardon the French loan word, but English needs it badly) portmanteau word.

Despite its wide coverage, though, even M-W does not , at least in its online version, get to the precise usage in question here.

You can have something happen to/for you, have something done, have done with something; you can be ‘had’ by a fraudster or by a class mate in a game of hee (US tag) on the playground. And when you've had enough of that, you can say you’ve ‘had it’ with hee. This uninflected language will never be fully accounted for by rules and recorded definitions, even in this age of electronic counting of occurrence. English is especially tricky in that way. In the case you raised, there are no obvious verbal or grammatical cues to provide a searchable pattern. At least, I should be interested and grateful if any ELU contributor can find a way of doing it.

Have I answered the question? Well, not quite. I should say ‘approved’ is a past passive ‘participle’, qualifying the noun ‘suggestions’, which is the object of ‘had’.