Are all the GETs/PUTs to the www.lightstreamer.com domain and push.lightstreamer.com domain done from the lsengine.html page?
actually, not.
most requests are made from ajax_frame.html (that comes from the pushserver).
If needed the client falls back using lsform.html . As the environment (ie the app loaded from file:///) is unexpected from our point of view I can't exclude that such fallback may happen in your case.

Alternatively, if I edit the hosts file, should I ONLY add www.lightstreamer.com and not push.lightstreamer.com so that the assets from the www.lightstreamer.com domain will be picked up from the www folder, but that requests to push.lightstreamer.com will go out to the remote server?
yes that's correct if you're going to use our online server and your local webserver.