I’ve checked your website, and I can see that your post and your page with the same name have the same ID.
If you try to edit page version, it redirects you to the post edit screen.
I created a new page copying the content from the post version and it doesn’t have the same ID and slug as the post version. You can check it at http://alternatifecs.com/uc-berkeley-sertfika-ve-master-karsilastirmasi-2/. Also, it is not shown on the blog page.
You will need to create one post for UC Berkley, and one page for UC Berkley and everything will be ok. This is a WordPress bug, not the theme one.
Also, the search function is a WordPress problem, it is not using any part of template. I noticed that if you write “UC Berkley”, the page will be found, but not with all the small letters.
I’ve never seen such a behavior. For all the bugs you mentioned here, you should try and ask for solutions on WordPress blog.