Is there any way to open Chrome at full screen and with navigation buttons?
Chrome does not have navigation bar while in full screen. This is by design. A lot of people petitioned this, but this feature hasn't been changed.
You can use hot keys to navigate back, forth, Ctrl + Tab to switch tabs, but you cannot navigate to a different URL.
You can try this Chrome extension: Full Screen Helper for Google Chrome
It provides back/forward buttons, address bar, and the opened tabs.