Howdy Britt,
I would love to help you with this. Of course I can not write and test all the code for you, but I can definitely point you in the right direction.
The datebox/calendar icon is the <div class=”list-date” /> . If you strip that and its children out the date box will go away. To replace it with a featured image just call the featured image function: tribe_event_featured_image() . You will probably want to wrap the featured image in a div of your own making, along with some CSS to get it the right width and add some margin.
Does that all make sense? Will that work for you? Please let me know.
Cheers!
– Brook