Desktop, laptop, mobile, iPad, home header-bg correct size?

To resolve issue on the ipad & mobile, you can add the following code to the Editor CSS:

@media (max-width:480px) {
   body {
          background-size: 100% 480px !important;
}
}
@media (max-width: 1024px) {
      body {
          background-size: 100% 740px !important;
}
}

– Also, I have checked your site on the Ipad and see that the btn on home page still be red color. However, if you still face the issue, you can add the following code to the Editor CSS:

@media(min-width:768px) and (max-width:979px) {
.btn-primary, .vimeoyoutubepopup_video {
    background: linear-gradient(to bottom, #EE3939 0%, #8D0007 100%) repeat scroll 0 0 rgba(0, 0, 0, 0) !important;
    border-color: #FD0A10 rgba(0, 0, 0, 0) #8D0007 !important;
    border-right: medium none rgba(0, 0, 0, 0) !important;
}
}

Hope this help !

 Regarding the btn on the iPad, could you kindly provide us a screenshot for further checking?  Are you customizing the code inside your site? Let’s us know once done, we will check your site again.
I think you can use the image size 1400 x 700 px.