trigger datalist options on click event of text box

Question 1

Perhaps a solution might be this (Fiddle)

<!DOCTYPE HTML>
<html>
<head>
<meta charset="utf-8">
<title></title>
<style>
input::-webkit-calendar-picker-indicator {
  display: none;/*remove default arrow in Chrome*/
}
.required:after {
    content: url(http://s25.postimg.org/6k40u5hcr/arrow.png);
    margin-left: -20px; 
    padding: .1em;
    pointer-events:none;/*make png cliccable*/
}
</style>
</head>
<body>
    <span class="required"><input id="team" list="bestTeam"></span>
     <datalist id="bestTeam">
       <option value="Inter">
       <option value="Milan">
       <option value="Juve">
       <option value="Roma">
     </datalist>
   </body>
</html>

First of all i removed the default arrow in chrome with input::-webkit-calendar-picker-indicator
in after element i set a small image in content property
to make the image clickable use pointer-events:none
This solution works on the latest versions of Chrome, firefox and IE
If this solution does not suit your needs, i apologize for making you lose time.

Question 2

That is a browser built-in feature that can't be modified. Also datalist isn't supported by Safari in versions lower than 12.1 nor lower versions of IE10

In order to accomplish this functionality use this polyfill: webshim. This will give feature support to Safari and IE- and convert the datalist into a regular ul list that you can then easily attach events to display/hide the list without loosing the autocomplete functionality.