You'd be better of spending your time getting mentions for the business in the local newspapers, blogs, etc with a link back to the site associated with the listing if building local authority is your goal. Adding social links to your GMB is not going to do anything for you, unless you want people to hit up your social pages.

Why do you want people to visit FB or whatever? If they have searched for something local then they are ready to pick up the phone. Your profile should be enough to do this. It’s like putting you social links on your website - pointless. You want people to come from FB not go to FB.

These are the things I do:

Schema
Linking to social profiles on website
Including website address on social profiles
I keep the social profiles active each week

But it feels random what Google is doing as I cannot consistently get the profiles to appear.

Are their any opportunities I am missing that could improve the chances of the profiles appearing?

Try removing the words "Join Us" and instead just facts.  Also try just an image vs an image with text.

Tough one. Let us know if others get rejected. Might just be a fluke. Maybe it's classified as irrelevant or off-topic. 

Possibly because of too much text overlay on the image? Not sure...

Well, looking at those categories, I'm fresh out of ideas and I'll have to call this either a bug or a test. Really weird!

I checked with some friends and no one has seen this before for auto dealers. It has been seen for malls and shopping centers though, which makes me wonder about your categories... What categories do you have set for the Lewis Ford of Hays listing?

I agree with Darren. I work with hundreds of Car Dealer GMBs and this is the first time i have seen this. Let us know if anything changes.