How to have multiple icons for multiple instances of the same program?
There are easier ways to switch windows/apps than using the Dock. Most native, is  by using the Mission Control. You can activate Mission Control several ways, on laptops it's usually F3 (without the Fn key), from Trackpad/Keyboard (check preferences) then select which app and window you want. You can also group windows by app: Mission Control > check "Group windows by application." Use two-finger scroll to zoom-in on specific app when grouped. 

Couple useful shortcuts:


⌘+Tab: Switch between apps, use w/ Shift(⇧) to go in other directions.
⌘+`: Switch between windows of current app, use w/ Shift(⇧) to go in other directions.


Check out Keyboard Preference Pane for other shortcuts, or customise any.

You should use spaces, if you already aren't, this will help you sort out apps and create a better workflow.

EDIT: I forgot to answer your big question. To have multiple icons for each window, though it's not recommended; it'll mess up the mission controls and confuse you, you'd have to do it through Terminal:
open -n -a Google\ Chrome; -n for new instance, -a specifies the name of app. You can also supply files after that so it'll open those files with the default or app supplied. Do man open to read more about it.