All I could think of is a permission problem... Can you paste the the 3rd screenshot with the error message again, the path is not completely visible.

Oh well, we are constantly trying to reduce the slack in Sling and make its performance on par with JCR; of course it's always a bit of overhead, but Sling offers some nice features which JCR cannot do; mostly because it's a fixed API we cannot extend anymore.

I am not sure if your screenshot shows the same component. The first screenshot shows a resource with the resourcetype "ProductTile2016" while the second shows a resource with the resourcetype "productTileReference".      

This error message is unrelated to the save operation (as it happens in the context of rendering that page).   This ERROR message is unrelated to your problem.

Why do you use Sling Distribution and not the replication API? It's not clear from your post if you need to distribute content from author to publish or from publish to publish.

Looks like you are using editable templates in your site. That means, that at rendering time the content of the template is merged with the content of the actual page (unlike static templates where the template is copied into the location where the resulting page is to be created). That means in the...

Oh sorry, overlooked the reference to Cloud Service. And yes, you are totally correct, that you cannot manage custom runmodes with AEM as a Cloud Service.    

Yes, that's the drawback of the situation. Your index has approx 280k entries, so you might need to go through all of them. The best way would be a custom index, but I know that in later versions a lot of indexes have been adjusted to avoid issues like this. So I am not sure if you want to introduce...

The message above is just a warning. If reaching 100k nodes it will terminate the query. You can increase that limit (see https://jackrabbit.apache.org/oak/docs/query/query-engine.html#slow-queries-and-read-limits), but I would recommend to stick with the default.

Yes, but you cannot designate a publish instance "to be the one" which executes these jobs, at least not with the tooling provided by AEM itself.    Of course you can create a new runmode "PublishWithSlingJobs", configure one of your publishs to have it and then start of the job based on that runmod...