This layout is occurring because all of the elements are h's which are block level elements. In order to get it aligned as you wish - you need to either position it relatevely to the left, insert it into one the headings.
I wuld also say that this is entirely the wrong use of multiple heading elements
I would use a regular list - or a DL and apply the icon as a :before element to the left. Using a dl -means that yuo can just keep adding awards (ie: a new dt and associated dd's) as well as more / less descriptive content (ie: dd's) per award.
I also added classes to the dt's to allow different icons to be shown for each type of award. And I added a description class to the DD's which allows specific styling of the second line versus date line of content per award.
UPDATE - as per OP's comments - regarding wanting spinning icons when using the :before approach. I have added two lines of CSS to achive this
-webkit-animation: fa-spin 2s infinite linear;
animation: fa-spin 2s infinite linear;
This was from @SatAj 's answer to this question:
Note that this animation will not necessarily work on older browsers suc as IE8, but most modern browsers should cope. Also - not all FA icons look good spinning IMO.
dl {margin-left: 40px}
dt {font-size: 1.6em;position:relative;}
dt:not(:first-of-type) {margin-top: 2em}
dt:before {
content: "";
font-family: FontAwesome;
left: -40px;
position:absolute;
top: 5px;
-webkit-animation: fa-spin 2s infinite linear;
animation: fa-spin 2s infinite linear;
}
dt.achievement:before {
content: "\f091";
}
dt.academic:before {
content: "\f19d";
}
dd {margin-left: 0}
dd.description {font-size: 1.3em;}
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/font-awesome/4.7.0/css/font-awesome.min.css"/>
<h2>Honors and Awards</h2>
<div class="col-sm-12 col-md-6 col-lg-5">
<dl>
<dt class="achievement">Boy Scouts of America Eagle Scout Rank</dt>
<dd class="description">The Highest Rank in Scouting</dd>
<dd>April 2014</dd>
<dt class="academic" >Principles Academic Award</dt>
<dd class="description">The greatest achievement in academia</dd>
<dd>June 2017</dd>
</dl>
</div>