In what directory does the Dolphin HD browser store its files? I already have root access and tried to use ES File Explorer to search for "dolphin" but to no avail.
Applications store their data in a folders named after their package. 
These folders are located in data/data.  

The easiest way to find out the package name is to look at the Play Store url. 

https://play.google.com/store/apps/details?id=mobi.mgeek.TunnyBrowser


The package name is the part that comes after id=.
You can see that Dolphin's package name is mobi.mgeek.TunnyBrowser, so the full path is:

/data/data/mobi.mgeek.TunnyBrowser