Any idea how I can align the language selector widget to the side of the existing buttons in header? Right now the selector is overlapping with my menu items.
You need to put your language selector inside your header widget area and place it in a separate div. And use such attributes as height and padding to push it down to be inline with your nav menu - which you are also going to have to move to the left using some css as well, in order to get both in a line.
Please contact WPML I recall that there is a function call to add the flags to the menu system. This function needs to be added to the bottom of the theme functions.php You can look it up at wpml. So give it a shot and try that and submit a support question at wpml. make sure in your question to you ask for the language selector or the flags only....