:not(:empty) CSS selector is not working?

Being a void element, an <input> element is considered empty by the HTML definition of "empty", since the content model of all void elements is always empty. So they will always match the :empty pseudo-class, whether or not they have a value. This is also why their value is represented by an attribute in the start tag, rather than text content within start and end tags.

Also, from the Selectors spec:

The :empty pseudo-class represents an element that has no children at all. In terms of the document tree, only element nodes and content nodes (such as DOM text nodes, CDATA nodes, and entity references) whose data has a non-zero length must be considered as affecting emptiness;

Consequently, input:not(:empty) will never match anything in a proper HTML document. (It would still work in a hypothetical XML document that defines an <input> element that can accept text or child elements.)

I don't think you can style empty <input> fields dynamically using just CSS (i.e. rules that apply whenever a field is empty, and don't once text is entered). You can select initially empty fields if they have an empty value attribute (input[value=""]) or lack the attribute altogether (input:not([value])), but that's about it.

It is possible with inline javascript onkeyup="this.setAttribute('value', this.value);" & input:not([value=""]):not(:focus):invalid

Demo: http://jsfiddle.net/mhsyfvv9/

input:not([value=""]):not(:focus):invalid{
  background-color: tomato;
}

<input 
  type="email" 
  value="" 
  placeholder="valid mail" 
  onchange="this.setAttribute('value', this.value);" />

You could try using :placeholder-shown...

input {
  padding: 10px 15px;
  font-size: 16px;
  border-radius: 5px;
  border: 2px solid lightblue;
  outline: 0;
  font-weight:bold;
  transition: border-color 200ms;
  font-family: sans-serif;
}

.validation {
  opacity: 0;
  font-size: 12px;
  font-family: sans-serif;
  color: crimson;
  transition: opacity;
}

input:required:valid {
  border-color: forestgreen;
}

input:required:invalid:not(:placeholder-shown) {
  border-color: crimson;
}

input:required:invalid:not(:placeholder-shown) + .validation {
  opacity: 1;
}

  

<input type="email" placeholder="e-mail" required>
<div class="validation">Not valid</span>

no great support though... caniuse