As Kris mentioned, with the addition of profiles the designer url has changed and you need to specify a profile in the URL. You can think of profiles as extension points of designer. A profile can specify it's own stencil set as well as functionality on how to load and store processes among other things.
Currently we do not have a jBPM-specific standalone profile. We working on that so check the blog (http://blog.athico.com/) and docs (https://hudson.jboss.org/hudson/job/jBPM5/lastSuccessfulBuild/artifact/target/jbpm-5.1-SNAPSHOT-docs-build/jbpm-docs/html/ch12.html) for updates as it will be added very soon.
Using the "default" profile provided by the designer out-of-the-box is not really suited as it stores the processes in their native json format rather than bpmn2.
Regarding the JDK question. Oryx designer does not target Java5 and does not do anything to enforce Java 5 compatibility. It targets Java 6.