The Connections plugin is really out of date. I do not know why WordPress is not telling you but there has been many updates since the 8.2.3 version you have installed. The current version is 8.5.27. Are you perhaps blocking updates to Connections for some reason?
Also, those PHP notices are because PHP notices have been turned on for your site. It is highly recommended that you have these notices turned of on production/live sites as the notices in come case can reveal sensitive info about your web server. See this blog post which offers advice on how to disable these notices.
Hope that helps, let me know.