Question: Is there a way or a program that marks the icon of a file with a badge, similar to how Dropbox marks synced files with a green checkmark, and files that are currently syncing blue?

I'd use that to mark videos I've watched in a series.

Answer: 
You can use Rating in Windows Explorer as work around. Mark \* as read.

From File's Properties, it allows me to do file Rating from (zero star ~ 5 stars). Since I'm not using the rating feature to really rate a file, instead I use it for the purpose of marking it as a read file, no matter what file type it was. I also can sort out those read vs. unread files if I want.
[See this image](https://i.stack.imgur.com/u7AqA.png)

Click, show the hidden files from the folder options and hide the file you already viewed. Since you enabled to show the hidden files from the folder options windows will show the hidden files by blurring it. This way you can mark the viewed files See the photo attached that the DNA and RNA file is blurred.