Here are a few quickly composed answers to your questions...
Why can't JavaEE libraries function without an application server?
The services provided by JavaEE (container managed transactions, container managed dependency injection, timer service, etc..) inherently involve JavaEE compliant Application Servers (for example: GlassFish, JBoss, WebSphere, etc...). Therefore the JavaEE libraries serve no purpose without such a container. "Why do I need something as massive as JBoss just to compile simple code to send an email?" You don't. There are ways to send an email without JavaEE... But if you want to do it the JavaEE way, you need a JavaEE container.
Why are JavaEE libraries not included with JavaSE download?
The same reason that many libraries aren't included: it would be overkill. Since you can't even use the JavaEE libraries without an application server, why bother to include them? JavaEE should be downloaded if and when a developer installs an application server and decides to use JavaEE.
Why are there so many JavaEE offerings?
Are there really "so many" JavaEE offerings? If so, please list some of them. More accurately I believe there are multiple implementations of the same APIs.
What can one do with JavaEE that they can't do without standard Java?
Lots. You can't rely on an application server to manage transactions or persistence contexts without JavaEE. You can't allow an application server to manage EJB dependency injection without JavaEE. You can't use an application managed timer service without JavaEE. The answer to this question should make the answer to the first question quite clear... Most of the services provided by JavaEE require a JavaEE container.
What can you do with JavaSE that you can't do with JavaEE?
Um... I don't know.
When does a developer decide they need JavaEE?
This question is completely subjective... But if you need any of the services provided by JavaEE, you start to think about it. If you don't know what JavaEE is... you probably don't need it.
When does a developer decide they do not need JavaEE?
See previous answer.
Why is JavaEE library version not in sync with JavaSE version?
Good question. I won't pretend to know how to answer it... But I would guess the answer is: "because they're not in sync".
