Zen Version 1.3.8a, brand new installation.
I have noted this behaviour previously on a few sites I run, some with older versions of Zen Cart, but have never had the opportunity to have a proper look at it and put it down to one or other of the plug-ins installed.
I just installed Zen from scratch on a new site and noted the same behaviour so went looking for a solution and came across this and other threads.
I am certain I am not speaking only of the legend at the top of the page and confirm that I see a red dot against all orders.
All these orders are for downloads.
I went through the process of checking product settings for the download products and ensured that they are NOT virtual products and that they are NOT free shipping. Still had the red dots.
I dug deeper and to cut a long story short, found that when an order is made for a download product, a space is inserted into the delivery_name field in the orders table.
The code responsible for showing the red dot checks whether or not the delivery and billing names are the same or whether the delivery_name field is empty.
As the billing name and delivery names are not the same (one is empty, the other is a space) and the delivery_name field has a space in it rather than being empty the red dot gets placed against the order.
I didn't manage to find the code responsible for writing to the db when a new order is made so for a quick and dirty fix I changed the code checking the names.
The code is in admin/orders.php around line 787 and I simply added a space between the single quotes at the end of the if statement ...['delivery_name'] != ' '))...
I ran a quick test and seems to work OK but someone might want to do a proper job rather than the hack I have done so things are stable for the future.
Hope this helps