I have no familiarity with ESEA at all, so I'll only speak in generalities about CrossOver.
First, CrossOver knows how to download and install Steam automatically. If you have downloaded it yourself, there's a point during CrossOver's installation process where you can tell it not to download but to instead use a local file.
If you were to double-click on an installer program in the Finder, CrossOver would run it but the built-in smarts about installation would not kick in. I would not recommend this.
The same things apply to an unsupported program such as ESEA. If a program requires installation, you should use CrossOver's Install Software feature, and use the Install Unsupported Software button. Make sure you're installing into the desired bottle. If you select Install Software from the Configure menu, you are targeting the default bottle. If you use the Install Software button from the Applications tab of the Manage Bottles window, you are targeting the bottle which is selected on the left side of that window.
Since ESEA is unsupported software, I don't know if it works. We provide a trial version of CrossOver so that you can find that out for yourself before buying. 😊