Those errors are not unique to mountain lion. You need a Java runtime installed to install any version of photoshop or elements, for one thing. Also see this:
and see if there's anything there that helps.
BTW, this isn't adobe support. It's a user to user forum.
After installing mountain lion I could not get my tool bar to appear. However it did appear in older versions of PE and they worked fine (contrary to common wisdom). I deleted PE (just the main app not all the system aspects), restarted and reinstalled and everything works.
Is anyone able to confirm if they've got it working OK in Mountain Lion? I don't want to upgrade if it's going to stop working.