Hi Urban Corps,
I was able to access your web map (http://arcg.is/14yDm1), open it in Collector and successfully added a new line and polygon - without any issues. I am using an iPhone 7.
You can check for yourself, please open the web map and search for: 380 New York St, Redlands, CA - you should see a new line (trail) and polygon (parking) features located on the north end of the Esri campus.
I also cannot see your referenced screenshots.
Hope this helps,