Hi Alan -
Essentially, anyone who views a Tableau dashboard/image served by Tableau Server needs to be licensed. This is the case regardless of whether the dashboard/image is viewed directly in Tableau Server or if you dynamically embed it into your web page.
You don't have to core licensing to embed - however, refer back to rule #1 - everyone who views the dashboard must be licensed. You can do this via named user (per user) licensing model or core licensing depending on what works best for you.
Embedded images are sometimes live, depending on how they are inserted into the web page and whether or not the image itself has been cached by your browser. In the example you point to, we will request the image "live" from the server if the browser doesn't already have it in cache. Just because one requests the image doesn't necessarily mean you'll get up-to-date data, though - you'd need to do something like add the :refresh=yes parameter to the URL you're building. If you don't, Tableau server may give you a new image, but it'll be based on potentially old, cached data on Tableau Server itself.
Question #4 is answered by the information you point to in question #3, unless I misunderstand what you're after.
Desktop and Server (core or named user) licenses are two different things. If you want to use Desktop, you buy a Desktop. If you then want to publish and/or consume reports on Server, you get a Server license. Having a Server license gives you no rights to use Desktop.
Hope this helps? If you have additional licensing-related questions, it's probably best if you give one of our handy-dandy-super-duper-helpful sales folks a call