Its been some time, all these things are happening on google homepage, it happens only when you go to google.com via redirection from your country specific google home page like google.co.in I observe this behavior in google chrome browser, so not sure this happens in other browser as well.
So, lets come to the point. If you go to google.co.in or any other country specific google search page and then click the link go to google.com as shown in the image below.
when you are redirected to google.com, you will see the following things will disappear – google search button, I’m Feeling Lucky and Lower options links like Advertisings Program, About Google etc.
All these Links options and buttons below the google search query box appears only when you move your mouse over it as shown in the image below, you can test the same.