a) Eclipse comes with it's own Java compiler, so if you are using the Java Runtime Environment, you won't run into many issues, except that extras like jarsigner and possibly the JRE source code may be missing. The best way to verify your installation is through your package manager; however, if you installed by some other means, careful directory observation can usually differentiate the two (see below).
b) JRE libraries are typically compiled without debugging symbols present. SDK libraries typically have them. Java debugging is done by starting the JVM with command line options which open a debugging port. The SDK talks to the JVM requesting breakpoints be set through this port. Later if you decide to step the JVM, it also sends the step / jump / rewind debugging commands through this port. The "debugging" symbols are actually JVM bytecode tables, which reference which line of source code is in effect starting at a particular bytecode instruction. This allows debuggers to associate the running bytecodes with line numbers in the original source code.
c) Verify it by (from the menu) Window->Preferences (on the selector column) expand "Java", expand "Build Path" under "Java", and select "ClassPath variables". You will see a few variables, including one called "JRE_SRC", which should point to the src.zip file containing the public facing JRE library source code. It is a good idea to verify JRE_LIB at the same time.
A JDK home directory typically contains a "bin" sub-directory and a "jre" sub-directory, so if you only see a "bin" sub-directory, odds are good you are in a JRE home directory. With this knowledge, hopefully you'll be able to figure the rest out.
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