How to make this splash page responsive

as I understood you have two parts header, one for register one for logo, in bootstrap 3 :

<div class="row">
    <div class="col-md-6 col-xs-12">logo here</div>
    <div class="col-md-6 col-xs-12">Register here here</div>
</div>

It will make your header responsive. and if you want that your image be responsive you can do like this in bootstrap 3:

<img src="https://wordpress.stackexchange.com/questions/210782/..." class="img-responsive" alt="Responsive image">

It will auto-resize base on browser witdh.

for background image I suggest you to use css cover property:

html { 
  background: url(images/bg.jpg) no-repeat center center fixed; 
  -webkit-background-size: cover;
  -moz-background-size: cover;
  -o-background-size: cover;
  background-size: cover;
}