Welcome to the forum.
At a very general level, one can run different versions of PrE and PSE on the same machine.
Unfortunately, and directly related to your question, I do not know what limitations will exist, regarding the functionality between the different versions. I just do not know how Organizer will be impacted, and also if one will have full functionality with SlideShows and transfers from PSE x to PrE y. Hopefully, others will know exactly and can give you details of what you might be loosing.
1- A forum participant tested that slideshow workflow from Photoshop Elements 7 to Premiere Elements 8 and verified that it worked for him.
2 - General Organizer use is probably an entirely different question. I have not attempted that myself and don't know anyone who has. It seems doubtful that a specific catalog file could be shared between Premiere Elements 8 and Photoshop Elements 7.
3 - Since you mention that you are interested in upgrading to Premiere Elements 8, I recommend that you read relevant FAQs from the FAQ forum here, starting with
All good to know. I knew that many have used different versions of PrE and PSE, but did not know what limitations might exist.
Thank you and Happy New Year,