Search field, there’s a setting to disable it in the Basic Settings of the customizer.”Show Top Search”
As for structure data problems… not sure why you are getting errors. I popped in a url to the testing tool for one of the posts on my demo site and it looks like things are working, just that I don’t have anything setup such as email verification or google+ for this, so it gives notice for that. Give me a link to one of your posts where I can test this myself…but you want to make sure you have your authorship setup correctly as well.
As for that code… you can try that if you want, but I seen where it doesn’t work for some people.
**By the way, when posting code, please highlight the code, then on the forum editor, click on the code button on the editor….I just did that to your code which puts it in the wrapper.