Event registration's 'Continue' button disappears when Payment Processor is changed
The missing button is there - it just has display set to none. It looks like there is a bunch of javascript on the page that hides various things, then reshows them - but I'm guessing behaves differently for different payment processors.

The issue looks to be the javascript is using .prop to reshow the element. 

      $('.crm-submit-buttons input').prop('disabled', false);


This function was added to jQuery in 1.6 but it looks like you are using jQuery 1.4.4. If you are able to use a more recent version of jQuery you should hopefully see the missing button load. There are a number of solutions documented here:

To easily update your jquery you can use this module to check if it resolves the issue. https://www.drupal.org/project/jquery_update