@Shareaholic
5. I think you're thinking of "Install Plugins" instead (page title), found via:
Dashboard >> Plugins >> Add new
If so, yes, that's how Shareaholic was installed on sites.
6. No, no custom PHP
7. Sorry, I cannot do any screen shares as the plugin has been removed. I notified Shareaholic so that they were aware of the issue. Sites have moved to other solutions.
Some sites have caching plugins such as W3 Total Cache, others use WP Super Cache, and others use no cache. Disabling cache does not fix the issue.