How do you trap the keypress or key down event on a DIV element (using jQuery)?
What is required to give the DIV element focus?
How do you trap the keypress or key down event on a DIV element (using jQuery)?
What is required to give the DIV element focus?
(1) Set the tabindex attribute:
<div id="mydiv" tabindex="0" />
(2) Bind to keydown:
 $('#mydiv').on('keydown', function(event) {
    //console.log(event.keyCode);
    switch(event.keyCode){
       //....your actions for the keys .....
    }
 });
To set the focus on start:
$(function() {
   $('#mydiv').focus();
});
To remove - if you don't like it - the div focus border, set outline: none in the CSS.
See the table of keycodes for more keyCode possibilities.
All of the code assuming you use jQuery.
#Here example on plain JS:
document.querySelector('#myDiv').addEventListener('keyup', function (e) {
  console.log(e.key)
})
#myDiv {
  outline: none;
}
<div 
  id="myDiv"
  tabindex="0"
>
  Press me and start typing
</div>
tabindex HTML attribute indicates if its element can be focused, and if/where it participates in sequential keyboard navigation (usually with the Tab key). Read MDN Web Docs for full reference.
$( "#division" ).keydown(function(evt) {
    evt = evt || window.event;
    console.log("keydown: " + evt.keyCode);
});
#division {
  width: 90px;
  height: 30px;
  background: lightgrey;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div id="division" tabindex="0"></div>
var el = document.getElementById("division");
el.onkeydown = function(evt) {
    evt = evt || window.event;
    console.log("keydown: " + evt.keyCode);
};
#division {
  width: 90px;
  height: 30px;
  background: lightgrey;
}
<div id="division" tabindex="0"></div>