Thanks so much for your post! There are a few things to address in your question:
First, we would need to see the list of settings you had toggled prior to opening the breakout rooms to confirm you had the option enabled to allow participants to choose their own room. Could you please provide that?
Second, if you do not assign any participants to any rooms, and instead allow them to choose their own, there will not be a popup notification for them. Instead, they should see a 'Breakout Rooms' button appear on their Zoom toolbar where they can select their room.
Third, by default, Zoom will not share your Zoom windows whatsoever when sharing your screen / desktop. This would be why they could not see your list on your screen, but could see the picture behind it. To enable that, you will need to toggle on 'Show Zoom windows during screen share' in your Zoom web portal, and then check the box called 'Show my Zoom Windows to other participants when I am screen sharing' in your Zoom application's settings > Share Screen settings. Please see this article for more details: https://support.zoom.us/hc/en-us/articles/360061383571-Showing-Zoom-windows-during-screen-share.
I hope this helps, please let me know if you have any more questions!