re: I actually don’t see the option to update the plugin on the plugins page.
Tile Plus did not receive in admin update support until version 4.0 (current is 4.1, with 5.0 forthcoming). So you would not see the update in the admin. You have to download it from your purchase history. Seeing as 3.0.1 was released 04/03/2013 the support license expired sometime in 2014 for you so you would have to renew (at a 30% discount) before updating. If you do update you have to add your support license key to the Connections : Settings admin page under the Licenses tab in order to have support for in admin updates.
Now, that said. I’m not entirely convinced updating will resolve the issue because pagination support is handled by the core Connections plugin, not the template, so that code if current. And the code that actually handles the logic of the pagination links is a core WordPress function. This is why I said this issue must be unique to your site.
Like I mentioned, I’m thinking this is a theme\plugin conflict. Something messing with the WordPress
$wp_query->query global var, something a theme/plugin should not touch directly but there are some many old and now obsolete tutorials out there that theme/plugins have copy/pasted that cause these weird edge case bugs.
Hope that explains updating and what I think the cause of the bug is and why more clearly.
re: Here’s a temp admin if you’d like to check it out yourself
I tried logging, but the account is still a subscriber account so I do not have access to anything, not even the directory page (has a separate pass).