Steven Zahm@ David
Which version of Connections did you install? The latest, 8.5.14, removed direct use of the core WordPress function get_query_var() because of other plugins and theme occasionally breaking it resulting in the error you are seeing. The latest version implements a wrapper function which checks to make sure the function is not broken before it is used and if it is broken fails back to an alternative method so Connections can keep doing its thing even though that part of WordPress is broken.
If you’d like to read the gory technical details here:
http://connections-pro.com/support/topic/fatal-error-in-query-php-line-28/#post-372136
You situation seems to be a little different. I’ve never seen this happen in the admin the Architect plugin must being doing something interesting. I suspect Architect plugin just might, maybe, cause you other troubles with other plugins though it might be as obvious. I only mention it so you can keep it in the back of your mind incase you run into other odd little issues that can not be explained.
Now, all this said, If you were installing the latest version and still receiving this error, I’ll need a copy of Architect so I can see what it is doing. I might have to add another other checks to make sure get_query_var() is not broken before trying to use it.