Drop down menu cuts off or appears behind div

Use a z-index for your drop down menu.

.submenu { z-index: 1; }

Also, as a side note, if you are using prefixed properties you should also use the spec property as well which should appear last after the vendor specific prefixes. Example:

-webkit-border-radius: 5px;
   -moz-border-radius: 5px;
        border-radius: 5px;

When a vendor supports the standard property, you don't have to change your code.

set the higher z-index of drop down menu .

.submenu { z-index: 999; }

reason "An element with greater stack order is always in front of an element with a lower stack order."