1-2) Java core is an apparently outdated name for what is called Lang and Util Base Libraries and Other Base Packages in this scheme . If you look at the Java Se 6 documentation, it refers to the list of Base Libraries , that is, you can assume that Java core == Base Libraries. But by and large this knowledge is of little use.
3) An interesting question, look for this scheme Java Fx is not part of the Java SE API, but is included in the JDK and in general is still part of the Jave SE Platform.
4) See Java EE is basically a set of interfaces that other providers implement. Java EE is not included in the JDK and you need to add these classes separately. In fact, the main criterion is not that Java EE applications are deployed in a servlet container or application server, but using Java EE interfaces. However, this is all very blurry, since JPA interfaces (on which Hibernate is based) or Dependency Injection does not use just lazy or Hello world applications. For good, clean Java SE applications are almost gone, on the other hand, real Java EE are huge monsters on weblogic or websphere. For simplicity, while you have a small application using Java EE slightly (like Hibernate), you can assume that you have Java SE with a small amount of EE as soon as it turns into an Enterprise monster - pure EE.
5) Here is a difficult question on the one hand Spring supports EE interfaces and is integrated with many EE solutions: JSP, Dependency injection standard, JPA, Hibernate, and more. Moreover, the project site says "Let's build a better Enterprise.", That is, first of all, this is an Enterprise solution. On the other hand, no one bothers to use Spring without bundles with EE technologies or a minimum of such technologies. Moreover, you can consider Spring as a competitor to Java EE, as a symbiosis with Java EE or as part of Java EE solutions, everything depends on each individual case. So it can be said that Spring is both an EE and SE solution, like a hammer that can be used to build a skyscraper, or it can be used to fix a barn.
6) Hmm, Java Android is a separate place in general, Google took the Java language specification and made its JVM and its language, it is generally compatible with the rest of Java, but not completely. Orakl long tried to sue Google for using Java. So Android is a completely different world, not very much connected with Java SE or EE.