How to have Chrome cover everything except the dock when double clicked?
Hold the Option key at the same time you click on the Maximize button on Chrome.  This will maximize the window while leaving visible the menu bar and dock.  This will also work on any other window on a Mac.

If this doesn't maximize the window on the first try, drag the window to where you want it (like you're doing now with the left and right sides) and then close the browser.  Next time you open it, it should fully maximize.