How can I count how many times a node appears in a view?
This can be done by tracking events with the help of Google Analytics 
Drupal provides module for it 
https://www.drupal.org/project/google_analytics

Events can be tracked by event tracking module
https://www.drupal.org/project/google_analytics_et

Will need to write custom js for making this event tracking, see documentation of google analytics event tracking module

I'm adding a code sample for tracking click on social media share icons: 

    $('.facebook_popup', context).click(function (e) {
    // First register an event in Google Analytics
    if (typeof ga != "undefined") {
      if (jQuery(e.target).closest('.node-social-media').length) {
        ga('send', 'event', 'social-side-bar', 'fb-share', window.location.pathname);
      }
      else if (jQuery(e.target).closest('#block-menu-menu-share').length) {
        ga('send', 'event', 'social-shares-top', 'fb-share', window.location.pathname);
      }
    }


This might help you creating google analytics events