Are Program Windows Self Controlled or OS Controlled?
 

Are third-party program windows self controlled or are they controlled by the OS?

I can't think of a better way to ask that so here is an example:

I've got Chrome open, when I click on an HTML file instead of opening the existing window, a new browser window will open. Is this Chrome's doing or is Linux telling it to open a new window?

I'm running Ubuntu 14.04 with Cinnamon desktop 2.6.12, but I feel like this is likely a general Linux question that would apply across distros/desktops.

that is chrome's doing.
however, it couldn't do so if the windowmanager wouldn't allow it.

As above it's the browser but not just the Chrome browser, it is also generally an option or choice of action on almost every browser.

Their actions are not at all similar to other programs you might use.

Chrome does it, but it does it by telling lower layers (including X) to open the window. The window manager decides where the window goes.

Maybe the OP was thinking that chrome program supplied it's window manager? Programs that run in a gui or window manager as we know it must be able to access the underlying gui be it x or gtk or gnome or cde or what not to run. A few can run a sort of gui in bash or other ways but not exactly like chrome does.