Kindly make sure you are installing the correct versions of ES and Kibana. Some versions don't work with others; I would suggest downloading the latest versions for each.
Also regarding your last message, going to localhost:9200 and seeing that means that ES is up and running. You may have closed your cluster, or Elasticsearch, and that is why you are unable to see that JSON anymore.
If you go to Elasticsearch's doc (on their website), you will be able to find clear and very straight forward steps to install both Kibana and ES. As far as I know, you do not need to use any other applications or even docker to get these two running together.
Also kindly note that if you were to install them both together, this means Kibana will be reading the data off of Elasticsearch, because that is the default settings for Kibana; it will be listening on port 9200.
As far as I know, you can't use Kibana with other indexers or search engines so simply. This also goes for embedding your Kibana dashboards/graphs. You will find within Kibana URLs to use in iframes, sure, but it wont be as reliable/customizable as you would image.
If I were you, before going into setting up Kibana and ES, I would first read the documentations on Elasticsearch's website. They are very, very, VERY helpful.