I understand you have a few questions about how the New Tab page works.
If you click on the small cogwheel icon in the top right corner, you will have the option to manage the content that is displayed on the page.
Currently you cannot change the layout of the New Tab page, but I recommend having a look at addons.mozilla.org where you may find an add-on that will help you customize that page.
If you hover the mouse pointer over a website tile, you will have the option to dismiss or delete the site from your history. You can also pin an existing site to its position so it will not change position as you browse the web.
For more information, see the Customize your Firefox New Tab page and Customize your New Tab page to hide or display content articles.
