Am I using the Java 7 try-with-resources correctly

It’s correct and there’s no requirement for catch clause. Oracle java 7 doc says the resource will be closed regardless of whether an exception is actually thrown or not.

You should use a catch clause only if you want to react upon the exception. The catch clause will be executed after the resource is closed.

Here’s a snippet from Oracle’s tutorial:

The following example reads the first line from a file. It uses an
instance of BufferedReader to read data from the file. BufferedReader
is a resource that must be closed after the program is finished with
it:

static String readFirstLineFromFile(String path) throws IOException {
    try (BufferedReader br =
                   new BufferedReader(new FileReader(path))) {
        return br.readLine();
    }
} // In this example, the resource declared in the try-with-resources statement is a BufferedReader.

… Because the BufferedReader instance is declared in a
try-with-resource statement, it will be closed regardless of whether
the try statement completes normally or abruptly (as a result of the
method BufferedReader.readLine throwing an IOException).

EDIT

Regarding the new edited question:

The code in Java 6 executes the catch and afterwards the finally block. This causes the resources to be still potentially opened in the catch block.

In Java 7 syntax, resources are closed before the catch block, so resources are already closed during the catch block execution. This is documented in the above link:

In a try-with-resources statement, any catch or finally block is run
after the resources declared have been closed.