I haven't been able to find an answer on whether or not this can happen. If there is a poorly built plugin that hasn't been updated in two years, and I make the poor decision to install it anyway - is it possible that even after complete deletion in the FTP this plugin could still be effecting my site? Has anyone ever dealt with something like this before and how did you fix it? I imagine I might have to check out the database if the plugin is causing this funky behavior, but I'm not sure where to start to rule this out.
Yes, a plugin that does not properly clean up after itself can continue to affect your site even after deletion. It can leave stuff behind in the database, or make alterations to your .htaccess file, which is probably the case in your scenario.
You might need to restore your .htaccess file - look here for help with that.